Chapter 5 Symmetry in Quantum Mechanics

Roughly speaking, a symmetry of some mathematical object is an invertible transformation that leaves all relevant structure as it is. Thus a symmetry of a set is just a bijection (as sets have no further structure, whence invertibility is the only demand on a symmetry), a symmetry of a topological space is a homeomorphism, a symmetry of a Banach space is a linear isometric isomorphism, and, crucially important for this chapter, a symmetry of a Hilbert space H is a unitary operator, i.e., a linear map u : H → H satisfying one and hence all of the following equivalent conditions: ∗ ∗ • uu = u u = 1H ; • u is invertible with u−1 = u∗; • u is a surjective isometry (or, if dim(H) < ∞, just an isometry); • u is invertible and preserves the inner product, i.e., uϕ,uψ = ϕ,ψ (ϕ,ψ ∈ H). The discussion of symmetries in quantum physics is based on the above idea, but the mathematically obvious choices need not be the physically relevant ones. Even in elementary quantum mechanics, where A = B(H), i.e., the C*-algebra of all bounded operators on some Hilbert space H, the concept of a symmetry is already diverse. The main structures whose symmetries we shall study in this chapter are:

1. The normal pure state space P1(H), i.e., the set of one-dimensional projections on H, with transition probability τ : P1(H)×P1(H) → [0,1] deﬁned by (2.44). 2. The normal state space D(H), i.e. the convex set of density operators ρ on H. 3. The self-adjoint operators B(H)sa on H, seen as a Jordan algebra (see below). 4. The effects E (H)=[0,1]B(H), seen as a convex partially ordered set (poset). 5. The projections P(H) on H, seen as an orthocomplemented lattice. 6. The unital commutative C*-subalgebras C (B(H)) of B(H), seen as a poset. Each of these structures comes with its own notion of a symmetry, but the main point of this chapter will be to show these notions are equivalent, corresponding in all cases to either unitary or—surprisingly—anti-unitary operators, both merely deﬁned up to a phase. The latter subtlety will open the world of projective unitary 1 group representation to quantum mechanics (without which the existence of spin- 2 particles such as electrons, and therewith also of ourselves, would be impossible).

© The Author(s) 2017 125 K. Landsman, Foundations of Quantum Theory, Fundamental Theories of Physics 188, DOI 10.1007/978-3-319-51777-3_5 126 5 Symmetry in quantum mechanics 5.1 Six basic mathematical structures of quantum mechanics

We ﬁrst recall the objects just described in a bit more detail. We have:

2 ∗ P1(H)={e ∈ B(H) | e = e = e,Tr (e)=dim(eH)=1}; (5.1) D(H)={ρ ∈ B(H) | ρ ≥ 0,Tr (ρ)=1}; (5.2) ∗ B(H)sa = {a ∈ B(H) | a = a}; (5.3)

E (H)={a ∈ B(H) | 0 ≤ a ≤ 1H }; (5.4) P(H)={e ∈ B(H) | e2 = e∗ = e}; (5.5)

C (B(H)) = {C ⊂ B(H) | C commutative C*-algebra,1H ∈ C}. (5.6)

The point is that each of these sets has some additional structure that defines what it means to be a symmetry of it, as we now spell out in detail. Definition 5.1. Let H be a Hilbert space (not necessarily finite-dimensional). 1. AWigner symmetry (of H) is a bijection

W : P1(H) → P1(H) (5.7)

that satisﬁes Tr (W(e)W( f )) = Tr (ef), e, f ∈ P1(H). (5.8) 2. A Kadison symmetry is an afﬁne bijection

K : D(H) → D(H), (5.9)

i.e. a bijection K that preserves convex sums: for t ∈ (0,1) and ρ1,ρ2 ∈ D(H),

K(tρ1 +(1 −t)ρ2)=tKρ1 +(1 −t)Kρ2. (5.10)

3. a. A Jordan symmetry is an invertible Jordan map

J : B(H)sa → B(H)sa, (5.11)

i.e., an R-linear bijection that satisﬁes the equivalent conditions

J(a ◦ b)=J(a) ◦ J(b); (5.12) J(a2)=J(a)2. (5.13)

Here ◦ = 1 ( + ) a b 2 ab ba (5.14)

is the Jordan product on B(H)sa, which turns the (real) vector space B(H)sa into a Jordan algebra,cf.§C.25. b. A weak Jordan symmetry is an invertible weak Jordan map, i.e., a bijection J ∈ C ( ( )) (5.11) of which the restriction |Csa is a Jordan map for each C B H . 5.1 Six basic mathematical structures of quantum mechanics 127

4. A Ludwig symmetry is an afﬁne order isomorphism

L : E (H) → E (H). (5.15)

5. A von Neumann symmetry is an order isomorphism

N : P(H) → P(H) (5.16)

preserving orthocomplementation, i.e. N(1 − e)=1 − N(e) for each e ∈ P(H). 6. A Bohr symmetry is an order isomorphism

B : C (B(H)) → C (B(H)). (5.17)

In nos. 3 and 5–6, an order isomorphism O of the given poset is a bijection that preserves the partial order ≤ (i.e., if x ≤ y, then O(x) ≤ O(y)) and whose inverse O−1 does so, too; cf. §D.1. The names in question have been chosen for historical reasons and (except perhaps for the ﬁrst and third) are not standard. Let us note that any Jordan map has a unique extension to a C-linear map

JC : B(H) → B(H); (5.18) ∗ ∗ JC(a )=JC(a) , (5.19) which satisﬁes (5.12) for all a,b, as well as

JC(a + ib)=J(a)+iJ(b), (5.20) with notation as in Proposition 2.6. Conversely, such a Jordan map (5.18) defines J = J a real Jordan map (5.11) by |B(H)sa . Similarly, a weak Jordan symmetry is equivalent to a map (5.18) that satisfies (5.19), preserves squares as in (5.13), and is linear on each subspace C of B(H), with C ∈ C (B(H)). In other words (in the spirit of Bohrification), JC is a homomorphism of C*-algebras on each commutative unital C*-subalgebra C ⊂ B(H). Therefore, either way J and JC are essentially the same thing, and if no confusion may arise we call it J. Note that a weak Jordan map J a priori satisfies (5.12) only for commuting self-adjoint a and b. It follows that weak (and hence ordinary) Jordan symmetries are unital: since

J(b)=J(1H ◦ b)=J(1H ) ◦ J(b) (5.21)

−1 for any b, we may pick b = J (1H ) to ﬁnd, reading (5.21) from right to left,

J(1H )=J(1H ) ◦ 1H = 1H . (5.22)

The special role of unitary operators u now emerges: each such operator deﬁnes the relevant symmetry in the obvious way, namely, in order of appearance: 128 5 Symmetry in quantum mechanics

W(e) = ueu∗; (5.23) K(ρ) = uρu∗; (5.24) L(a) = uau∗; (5.25) J(a) = uau∗; (5.26) N(e) = ueu∗; (5.27) B(C) = uCu∗, (5.28)

∗ where a = a in (5.26). If not, this formula remains valid also for the map JC. Fur- thermore, in (5.28) the notation uCu∗ is shorthand for the set {uau∗ | a ∈ C}, which is easily seen to be a member of C (B(H)). Here, as well as in the other three cases, it is easy to verify that the right-hand side belongs to the required set, that is,

∗ ∗ ∗ ueu ∈ P1(H), uρu ∈ D(H), uρu ∈ E (H), (5.29) ∗ ∗ ∗ uau ∈ B(H)sa, uρu ∈ P(H), uCu ∈ C (B(H)), (5.30) respectively, provided, of course, that

e ∈ P1(H), ρ ∈ D(H), a ∈ E (H) a ∈ B(H)sa, e ∈ P(H), C ∈ C (B(H)).

Indeed, if, in (5.23), e = eψ = |ψihψ| for some unit vector ψ ∈ H, then

∗ ueψ u = euψ . (5.31)

If ρ ≥ 0 in that hψ,ρψi ≥ 0 for each ψ ∈ H, then clearly also uρu∗ ≥ 0, and if Tr(ρ) = 1, then also Tr(uρu∗) = 1. If a∗ = a, then

(uau∗)∗ = u∗∗a∗u∗ = uau∗. (5.32)

However, one may also choose u in these formulae to be anti-unitary, as follows: Deﬁnition 5.2. 1. A real-linear operator u : H → H is anti-linear if

u(zψ) = zψ (z ∈ C). (5.33)

2. An anti-linear operator u : H → H is anti-unitary if it is invertible, and

huϕ,uψi = hϕ,ψi (ϕ,ψ ∈ H). (5.34) The adjoint u∗ of a (bounded) anti-linear operator u is deﬁned by the property

hu∗ϕ,ψi = hϕ,uψi (ϕ,ψ ∈ H), (5.35) in which case u∗ is anti-linear, too. Hence we may equally well say that an anti-linear ∗ ∗ operator is anti-unitary if uu = u u = 1H . The simplest example is the map 5.1 Six basic mathematical structures of quantum mechanics 129

J : Cn → Cn; Jz = z, (5.36)

n i.e., if z =(z1,...,zn) ∈ C , then (Jz)i = zi. Similarly, one may deﬁne

J : 2 → 2; Jψ = ψ, (5.37) and likewise on L2, where complex conjugation is deﬁned pointwise, that is,

(Jψ)(x)=ψ(x). (5.38)

For any Hilbert space one may pick a basis (υi) and deﬁne J relative to this basis by

J ∑ciυi = ∑ciυi. (5.39) i i

For future use, we state two obvious facts. Proposition 5.3. 1. The product of two anti-unitary operators is unitary. 2. Any anti-unitary operator u : H → H takes the form u = Jv, where v is unitary and J is an anti-unitary operator on H of the kind constructed above. It is an easy verification that (5.23) - (5.28) still define symmetries if u is anti- unitary. Note that in terms of the complexification JC, eq. (5.26) should read

∗ ∗ JC(a)=ua u . (5.40)

The goal of the following sections is to show that these are the only possibilities: Theorem 5.4. Let H be a Hilbert space, with dim(H) > 1. 1. Each Wigner symmetry takes the form (5.23); 2. Each Kadison symmetry takes the form (5.24); 3. Each Ludwig symmetry takes the form (5.25); 4. a. Each Jordan symmetry takes the form (5.26); b. If dim(H) > 2, also each weak Jordan symmetry takes this form; 5. If dim(H) > 2, each von Neumann symmetry takes the form (5.27); 6. Again if dim(H) > 2, each Bohr symmetry takes the form (5.28), where in all cases the operator u is either unitary or anti-unitary, and is uniquely determined by the symmetry in question up to a phase (that is, u and u implement the same symmetry by conjugation iff u = zu, where z ∈ T). As we shall see, the reason why the case H = C2 is exceptional with regard to weak Jordan symmetries, von Neumann symmetries, and Bohr symmetries is that in those cases the proof relies on Gleason’s Theorem, which fails for H = C2. To see this more explicitly, and also to prove the positive cases (i.e., nos. 1–4a) in a simple situation without invoking higher principles, before proving Theorem 5.4 in general it is instructive to ﬁrst illustrate it in the two-dimensional case H = C2. 130 5 Symmetry in quantum mechanics 5.2 The case H = C2

We start with some background. Any complex 2 × 2 matrix a can be written as

3 = ( , , , )= 1 σ ( ∈ C) a a x0 x1 x2 x3 2 ∑ xμ μ xμ ; (5.41) μ= 0 10 01 0 −i 10 σ = , σ = , σ = , σ = , (5.42) 0 01 1 10 2 i 0 3 0 −1 i.e., the Pauli matrices. Furthermore, if we equip the vector space M2(C) of complex 2 × 2 matrices√ with the canonical inner product (2.34), then the rescaled matrices

σμ = σμ / 2 form a basis (≡ orthonormal basis) of the ensuing Hilbert space. Writing x =(x1,x2,x3), some interesting special cases are: • ∈ R = ∈ R3 2 + 2 + 2 + 2 = ∈ ( ) x0 , x iv with v and x0 v1 v2 v3 1, which holds iff a SU 2 ; ∗ • xμ ∈ R for each μ = 0,1,2,3, which is the case iff a = a. 3 • x0 = 1, x ∈ R , and x = 1, which holds iff a is a one-dimensional projection. The ﬁrst case follows because SU (2) consist of all matrices of the form αβ , α,β ∈ C, |α|2 + |β|2 = . −β α 1 (5.43)

The second case is obvious, and the third follows from Proposition 2.9. Assume the third case, so that a = e with e2 = e∗ = e and Tr(e)=1. If a linear map u : C2 → C2 is unitary, then simple computations show that e = ueu∗ is a one- = 1 ∑3 σ = ∈ R3 dimensional projection, too, given by e 2 μ=0 xμ μ with x0 1, x , and x = 1. Writing x = Rx for some map R : S2 → S2,wehave

u(x · σ)u∗ =(Rx) · σ, (5.44)

· σ = 3 σ where x ∑ j=1 x j j. This also shows that R extends to a linear isometry R : 3 3 R → R . Using the formula Tr(σiσ j)=2δij, the matrix-form of R follows as

= 1 ( σ ∗σ ). Rij 2 Tr u iu j (5.45) Deﬁne U(2) as the (connected) group of all unitary 2×2 matrices (whose connected subgroup SU (2) of elements with unit determinant has just been mentioned). Also, recall that O(3) is the group of all real orthogonal 3×3 matrices M, a condition that may be expressed in (at least) four equivalent ways (like unitarity): T M • MM = M M = 13; • M invertible and MT = M−1; • M is an isometry (and hence it is injective and therefore invertible); • M preserves the inner product: Mx,My = x,y for all x,y ∈ R3. 5.2 The case H = C2 131

This implies det(M)=±1 (as can be seen by diagonalizing M; being a real linear isometry, its eigenvalues can only be ±1, and det(M) is their product). Thus O(3) breaks up into two parts O±(3)={R ∈ O(3) | det(R)=±1}, of which O+ ≡ SO(3) consists of rotations. Using an explicit parametrization of SO(3), e.g., through Euler angles, or, using surjectivity of the exponential map (from the Lie algebra of SO(3), which consist of anti-symmetric real matrices), it follows that O±(3) are precisely the two connected components of O(3), the identity of course lying in O+(3). Proposition 5.5. The map u → R deﬁned by (5.44) is a homomorphism from U(2) onto SO(3). In terms of SU(2) ⊂ U(2), this map restricts to a two-fold covering

π˜ : SU (2) → SO(3), (5.46) with discrete kernel ker(π˜)={12,−12}. (5.47) Proof. As a ﬁnite-dimensional linear isometry, R is automatically invertible (this also follows from unitarity and hence invertibility of u), hence R ∈ O(3).Itisob- vious from (5.44) that u → R is a continuous homomorphism (of groups). Since U(2) is connected and u → R is continuous, R must lie in the connected component of O(3) containing the identity, whence R ∈ SO(3). To show surjectivity of π˜, take ∈ R3 = ( 1 θ)+ ( 1 θ) ·σ some unit vector u and deﬁne u cos 2 isin 2 u . The corresponding rotation Rθ (u) is the one around u by an angle θ, and such rotations generate SO(3). Finally, it follows from (5.44) that u ∈ ker(π˜) iff u commutes with each σi and hence, by (5.41), with all matrices. Therefore, u = z·12 for some z ∈ C, upon which the the condition det(u)=1 (in that u ∈ SU (2)) enforces z = ±1.

Note that the covering (5.46) is topologically nontrivial (i.e., SU (2) = SO(3)×Z2), since SU (2) =∼ S3 is simply connected, whereas SO(3) is doubly connected: a closed path t → R2πt (u), t ∈ [0,1] in SO(3) (starting and ending at 13) lifts to a path

t → cos(πt)+isin(πt)u · σ in SU (2) that starts at the unit matrix 12 and ends at −12. To incorporate O−(3), let Ua(2) be the set of all anti-unitary 2 × 2 matrices. These do not form a group, as the product of two anti-unitaries is unitary, but the union U(2) ∪Ua(2) is a disconnected Lie group with identity component U(2). Proposition 5.6. The map u → R deﬁned by (5.44) is a surjective homomorphism

π˜ : U(2) ∪Ua(2) → O(3), (5.48)

with kernel U(1), seen as the diagonal matrices z · 12,z∈ T. Moreover, π˜ maps U(2) onto SO(3) and maps Ua(2) onto O−(3). Proof. The map u → R in (5.44) sends the anti-unitary operator u = J on C2 to R = diag(1,−1,1) ∈ O−(3). Since Ua(2)=J ·U(2) and similarly O−(3)=R·SO(3), the last claim follows. The computation of the kernel may now be restricted to U(2), and then follows as in the last step op the proof of the previous proposition. 132 5 Symmetry in quantum mechanics

We now return to Theorem 5.4 and go through its special cases one by one. Part 1 of Theorem 5.4 is Wigner’s Theorem, which in the case at hands reads: 2 2 Theorem 5.7. Each bijection W : P1(C ) → P1(C ) that satisﬁes

Tr (W(e)W( f )) = Tr (ef) (5.49)

2 ∗ for each e, f ∈ P1(C ) takes the form W(e)=ueu , where u is either unitary or anti-unitary, and is uniquely determined by W up to a phase. To prove, this we transfer the whole situation to the two-sphere, where it is easy: 2 Proposition 5.8. The pure state space P1(C ) corresponds bijectively to the sphere

S2 = {(x,y,z) ∈ R3 | x2 + y2 + z2 = 1},

2 in that each one-dimensional projection e ∈ P1(C ) may be expressed uniquely as 1 + zx− iy e(x,y,z)= 1 , (5.50) 2 x + iy 1 − z where (x,y,z) ∈ R3 and x2 + y2 + z2 = 1. Under the ensuing bijection

2 ∼ 2 P1(C ) = S , (5.51)

Wigner symmetries W of C2 turn into orthogonal maps R ∈ O(3), restricted to S2. Proof. The ﬁrst claim restates Proposition 2.9. If ψ and ψ are unit vectors in C2 with corresponding one-dimensional projections eψ (x,y,z) and eψ (x ,y ,z ) then, as one easily veriﬁes, the corresponding transition probability takes the form

( )= 1 ( + , )= 2( 1 θ( , )), Tr eψ eψ 2 1 x x cos 2 x y (5.52) where θ(x,y) is the arc (i.e., geodesic) distance between x and y. Consequently, 2 2 2 2 if W : P1(C ) → P1(C ) satisfies (5.8), then the corresponding map R : S → S 2 ∼ 2 (defined through the above identification P1(C ) = S ) satisfies

R(x),R(x ) = x,x (x,x ∈ S2). (5.53)

Lemma 5.9. If some bijection R : S2 → S2 satisﬁes (5.53), then R extends (uniquely) to an orthogonal linear map (for simplicity also called) R : R3 → R3.

3 Proof. With (u1,u2,u3) the standard basis of R , deﬁne a 3 × 3 matrix by

Rkl = uk,R(ul). (5.54)

−1 −1 It follows from (5.53) that R (u j)k = R jk, which implies R (u j),x = ∑k R jkxk, or, once again using (5.53), R(x) j = ∑k R jkxk. Hence the map x → ∑ j,k R jkxku j, i.e., the usual linear map deﬁned by the matrix (5.54), extends the given bijection R. Orthogonality of this linear map is, of course, equivalent to (5.53). 5.2 The case H = C2 133

Wigner’s Theorem then follows by combining Propositions 5.6 and 5.8: given the linear map R just constructed, read (5.44) from right to left, where u exists by surjectivity of the map (5.48), and the precise lack of uniqueness of u as claimed in Theorem 5.4 is just a restatement of the fact that (5.48) has U(1) as its kernel.

Kadison’s Theorem is part 2 of Theorem 5.4. Explicitly, for H = C2 we have: Theorem 5.10. Each afﬁne bijection K : D(C2) → D(C2) is given as K(ρ)=uρu∗, where u is unitary or anti-unitary, and is uniquely determined by K up to a phase.

Proof. We once again invoke Proposition 2.9, implying that any density matrix ρ on C2 takes the form 3 ρ = 1 + σ , 2 12 ∑ xμ μ (5.55) μ=1 with x≤1. Moreover, the ensuing bijection D(C2) =∼ B3, ρ → x, is clearly afﬁne, in that a convex sums tρ +(1 −t)ρ of density matrices correspond to convex sums tx +(1 −t)x of the corresponding vectors in R3. Lemma 5.11. Any afﬁne bijection K of the unit ball B3 in R3 is given by an orthogonal linear map R ∈ O(3).

3 2 Proof. First, K must map the boundary ∂eB = S to itself (necessarily bijectively): if x ∈ S2 and K(x)=tx +(1 −t)x , then x = tK−1(x )+(1 −t)K−1(x ), whence

K−1(x )=K−1(x ), (5.56) since x is pure, whence x = x , so that also K(x) is pure. Second, the basis of all further steps is the property

K(0)=0. (5.57)

This is because 0 is intrinsic to the convex structure of B3: it is the unique point ∈ 2 1 + 1 = with the property that for any x S there exists a unique x such that 2 x 2 x 0, namely x = −x. Thus 0 must be preserved under affine bijections. For a formal proof (by contradiction), suppose K(0) = 0, and define y = K(0)/K(0)∈S2. Then K(0) has an extremal decomposition K(0)=ty +(1 − t)y , with y = −y and t = 1 ( + K( )) K−1 2 1 0 . Applying the affine map then gives 1 + K(0) K−1(y ) = K−1(y)· . 1 −K(0)

Now y ∈ S2 and hence K−1(y) ∈ S2 by part one of this proof (applied to K−1), so that K−1(y) = 1. But this implies K−1(y ) > 1, which is impossible because y ∈ S2 and hence K−1(y ) = 1. Third, for x ∈ B3 and t ∈ [0,1] the preceding point implies that

K(tx)=K(tx +(1 −t)0)=tK(x)+(1 −t)K(0)=tK(x). (5.58) 134 5 Symmetry in quantum mechanics

The same then holds for x ∈ B3 and all t ≥ 0 as long as tx ∈ B3: for take t > 1, so that t−1 ∈ (0,1), and use the previous step with x tx and t t−1 to compute

K(tx)=tt−1K(tx)=tK(t−1tx)=tK(x).

Also, (5.58) and afﬁnity imply that for any x,y ∈ B3 for which x + y ∈ B3,wehave

K( + )= K( 1 + 1 )= · ( 1 K( )+ 1 K( )) = K( )+K( ). x y 2 2 x 2 y 2 2 x 2 y x y (5.59) With our earlier result (5.57), this also gives

K(−x)=−K(x). (5.60)

For some nonzero x ∈ R3, take s ≥x and t ≥x. Then by (5.58) we have t x sK(x/s)=sK = tK(x/t). s t We may therefore deﬁne a map R : R3 → R3 by

R(0)=0; (5.61) R(x)=s · K(x/s)(x = 0), (5.62) for any choice of s ≥x.Forx ∈ B3 we may take s = 1, so that R extends K. To prove that R is linear, for x ∈ R3 and t ≥ 0 pick some s ≥ tx and compute t t x t x R(tx)=sK x = sK x = s ·x K = tR(x). (5.63) s s x s x

For t < 0, we ﬁrst show from (5.60) and (5.62) that

R(−x)=−R(x), (5.64) upon which (5.63) gives

R(tx)=R(|t|·(−x)) = |t|R(−x)=−|t|R(x)=−tR(x). (5.65)

Furthermore, for given x,y ∈ B3, pick s > 0 such that s ≥x and s ≥y, so that s = 2s ≥x + y by the triangle inequality, and use (5.59) to compute x + y x y R(x + y)=sK = sK + = sK(x/s)+sK(y/s) s s s = R(x)+R(y). (5.66)

Finally, R is an isometry by (5.62) and step one of the proof. Being also linear and invertible, R must therefore be an orthogonal transformation. 5.2 The case H = C2 135

Given step one, an alternative proof derives this lemma from Proposition 5.18 below, which shows that the transition probabilities (5.52) on S2 are determined by the convex structure of B3, so that affine bijections must preserve them. In other words, the boundary map S2 → S2 defined by K preserves transition probabilities and hence satisfies the conditions of Lemma 5.9. This reasoning effectively reduces Kadison’s Theorem to Wigner’s Theorem, a move we will later examine in general. In any case, Theorem 5.10 now follows from Lemma 5.11 is exactly the same way as Theorem 5.7 followed from the corresponding Lemma 5.9.

We have given this proof in some detail, because step 3 will recur on other occasions where a given affine bijection is to be extended to some linear map. Ludwig’s Theorem is part 3 of Theorem 5.4. For H = C2,wehave: Theorem 5.12. Each affine order isomorphism L : E (C2) → E (C2) reads L(a)= uau∗, where u is unitary or anti-unitary, and is uniquely fixed by L up to a phase.

2 Proof. Using the parametrization (5.41), we have a(x0,x1,x2,x3) ∈ E (C ) iff each xμ is real and 0 ≤ x0 ±x≤2. In particular, we have 0 ≤ x0 ≤ 2. This easily follows from (2.38), noting that a ∈ E (C2) just means that a∗ = a and that both eigenvalues of a lie in [0,1]. Thus E (C2) is isomorphic as a convex set to a convex subset C of R4 [ , ] that is fibered over the x0-interval 0 2 , where the fiber Cx0 of C over x0 is the three-ball B3 with radius x = x as long as 0 ≤ x ≤ 1, whereas for 1 ≤ x ≤ 2 x0 0 0 0 the fiber is B3 ,soatx = 1 the fiber is C = B3 ≡ B3 (in one dimension less, 2−x0 0 1 1 this convex body is easily visualizable as a double cone in R3, where the fibers are disks). The partial order on C induced from the one on E (C2) is given by ( , ) ≤ ( , ) − ≥ − , x0 x x0 x iff x0 x0 x x (5.67) which follows from (5.41) and (2.38), noting that for matrices one has a ≤ a iff a − a has positive eigenvalues. A similar argument to the one proving (5.57) then shows that any affine bijection L of C must map the base space [0,2] to itself (as an affine bijection), and hence either x0 → x0 or x0 → 2 − x0. The latter fails to L L preserve order, so must fix x0. Similarly, maps each three-ball Cx0 to itself by an affine bijection, which, by the same proof as for Kadison’s Theorem above, must be induced by some element Rx of O(3). Finally, the order-preserving condition 0 x −x0 ≥x −x⇒x −x0 ≥R x −Rx x obtained from (5.67) and the property 0 0 x0 0 L( )= x0 x0 just found can only be met if Rx0 is independent of x0. Part 3 of Theorem 5.4 does not carry an official name; it may be attributed to Kadi- son, too, but the hard part of the proof was given earlier by Jacobson and Rickart. Rather than a contrived (though historically justified) name like “Jacobson–Rickart– Kadison Theorem”, we will simply speak of Jordan’s Theorem (for H = C2):

Theorem 5.13. Each linear bijection J : M2(C)sa → M2(C)sa that satisﬁes (5.13) and hence (5.12) takes the form J(a)=uau∗, where u is either unitary or anti- unitary, and is uniquely determined by J up to a phase. 136 5 Symmetry in quantum mechanics

Proof. First, any Jordan map (and hence a fortiori any Jordan automorphism) trivially maps projections into projections, as it preserves the deﬁning conditions e2 = e∗ = e. Second, any Jordan automorphism J maps one-dimensional projections into one-dimensional projections: if e ∈ P1(H), then J(e) = 0 and J(e) = 12, both because J is injective in combination with J(0)=0 and J(12)=12, respectively. Hence J(e) ∈ P1(H), since this is the only remaining possibility (a more sophisti- cated argument shows that this is even true for any Hilbert space H). From (5.41) and subsequent text, as in (5.44), by linearity of J we therefore have 3 3 J ∑ x jσ j = ∑(Rx) jσ j, (5.68) j=1 j=1 from some map R : S2 → S2, which is bijective because J is. Linearity of J then allows us to extend R to a linear map R3 → R3, with matrix

3 = 1 (σ J(σ )), R jk 2 ∑ Tr k j (5.69) j=1 cf. (5.45). By (5.69), this linear map restricts to the given bijection R : S2 → S2, which also shows that it is isometric. Thus we have a linear isometry on R3, which therefore lies in O(3). The proof may then be completed as in Theorem 5.7. The case H = C2 was already exceptional in the context of Gleason’s Theorem, and it remains so as far as weak Jordan symmetries and Bohr symmetries are concerned. 2 Proposition 5.14. The poset C (M2(C)) is isomorphic to {⊥}∪RP , where the real projective plane RP2 is the quotient S2/ ∼ under the equivalence relation x ∼−x, and the only nontrivial ordering is ⊥≤p for any p ∈ RP2. ∗ Proof. It is elementary that M2(C) has a single one-dimensional unital -subalgebra, namely C · 1, the multiples of the unit; this gives the singleton ⊥ in C (M2(C)). ∗ Furthermore, any two-dimensional unital -subalgebra C of M2(C) is generated by a one-dimensional projection e, in that C is the linear span of e and 12. Hence C is also the linear span of (the projection) 12 − e and 12. In our parametrization of all one-dimensional projections e on C2 by S2 (cf. Proposition 2.9), if e corresponds to 2 x, then 1 − e corresponds to −x. This yields the remainder RP of C (M2(C)). ∗ Finally, commutative unital -subalgebras D of M2(C) of dimension > 2 do not exist. For any such algebra D would contain some two-dimensional C just deﬁned, but a simple computation (for example, in a basis were C consists of all diagonal matrices) shows that the only matrices that commute with all elements of C already lie in C (i.e., are diagonal). Hence no commutative extension of C exists. Bohr symmetries B for C2 therefore correspond to bijections of RP2. Similarly, weak Jordan symmetries J for C2 corresponds to bijections of S2 (the difference with Bohr symmetries lies in the fact that J may also map C = span(e,12) to itself nontrivially, i.e., by sending e to 12 − e, which for B would yield the identity map). In both cases, few of these bijections are (anti-) unitarily implemented. 5.3 Equivalence between the six symmetry theorems 137 5.3 Equivalence between the six symmetry theorems

If dim(H) > 1, the first three claims of Theorem 5.4 are equivalent; if dim(H) > 2, all claims are. We will show this in some detail, if only because the proofs of the various equivalences relate the six symmetry concepts stated in Definition 5.1 in an instructive way. We will do this in the sequence Wigner ↔ Kadison ↔ Jordan, and subsequently Jordan ↔ Ludwig, Jordan ↔ von Neumann, and Jordan ↔ Bohr. Consequently, in principle only one part of Theorem 5.4 requires a proof. Although redundant, we will, in fact, prove both Wigner’s Theorem and Jordan’s (indeed, no independent proof of the other parts of Theorem 5.4 seems to be known!). The most transparent way to state the various equivalences is to note that in each case the set of symmetries of some given kind (i.e., Wigner, ...)forms a group. In all cases, the nontrivial part of the proof is the establishment of a “natural” bijection, from which the group homomorphism property is trivial (and hence will not be proved). Proposition 5.15. There is an isomorphism of groups between: • The group of affine bijections K : D(H) → D(H); • The group of bijections W : P1(H) → P1(H) that satisfy (5.8), viz.

W = K|P ( ); (5.70) 1 H K λ = λ W(υ ), ∑ ieυi ∑ i υi (5.71) i i

ρ = ∑ λ ρ ∈ D( ) where i ieυi is some (not necessarily unique) expansion of H in terms of a basis of eigenvector υi with eigenvalues λi, where λi ≥ 0 and ∑i λi = 1.In particular, (5.70) and (5.71) are well deﬁned.

Proof. It is conceptually important to distinguish between B(H)sa as a Banach space in the usual operator norm ·, and B1(H)sa, the Banach space of trace-class operators in its intrinsic norm ·1. Of course, if dim(H) < ∞, then B(H)sa = B1(H)sa as vector spaces, but even in that case the two norms do not coincide (although they are equivalent). The proof below has the additional advantage of immediately generalizing to the inﬁnite-dimensional case. We start with (5.70).

1. Since P1(H)=∂eD(H), by the same argument as in the proof of Lemma 5.11, any affine bijection of the convex set D(H) must preserve its boundary, so that K maps P1(H) into itself, necessarily bijectively. The goal of the next two steps is to prove that (5.70) satisfies (5.8), i.e., preserves transition probabilities. 2. An affine bijection K : D(H) → D(H) extends to an isometric isomorphism K1 : B1(H)sa → B1(H)sa with respect to the trace-norm ·1, as follows:

a. Put K1(0)=0 and for b ≥ 0, b ∈ B1(H), i.e. b ∈ B1(H)+, and b = 0, deﬁne

K1(b)=b1K(b/b1). (5.72) 138 5 Symmetry in quantum mechanics

By construction, K1 is isometric and preserves positivity. For b ∈ B1(H)+ we have Tr(b)=b1, hence b/b1 ∈ D(H), on which K is defined. Linearity of K1 with positive coefficients (as a consequence of the affine property of K) is verified as in the proof of Lemma 5.11; this time, use a b a + b =(a1 + b1) · t +(1 −t) , (5.73) a1 b1

with t = a1/(a1 +b1). Note that if a,b ∈ B1(H)+, then a+b ∈ B1(H)+. b. For b ∈ B1(H)sa, decompose b = b+ − b−, where b± ≥ 0; see Proposition A.24 (this remains valid in general Hilbert spaces). We then deﬁne

K1(b)=K1(b+) − K1(b−). (5.74)

To show that this makes K1 linear on all of B1(H)sa, suppose b = b+ − b−

with b± ≥ 0. Then b+ + b− = b+ + b−, and since each term is positive,

K1(b+ + b−)=K1(b+)+K1(b−)=K(b+ + b−)=K1(b+)+K1(b−),

by the previous step. Hence K1(b+) − K1(b−)=K1(b+) − K1(b−), so that (5.74) is actually independent of the choice of the decomposition of b as long as the operators are positive. Hence for a,b ∈ B1(H)sa we may compute

K1(a + b)=K1(a+ + b+ − (a− + b−)) = K1(a+ + b+) − K1(a− + b−)

= K1(a+)+K1(b+) − K1(a−) − K1(b−)=K1(a)+K1((b),

since a+ + b+ and a− + b− are both positive.

The key point in verifying isometry of K1 is the property |b| = b+ + b−, which follows from (A.76) or Theorem B.94. Using this property, we have

K1(b)1 = Tr (|K1b|)=Tr (|K1(b+) − K1(b−)|)=Tr (K1(b+)+K1(b−))

= Tr (b+ + b−)=Tr (|b+ − b−|)=Tr (|b|)=b1.

3. For any two unit vectors ψ,ϕ in H we have the formula eψ − eϕ 1 = 2 1 − Tr (eψ eϕ ), (5.75)

which can easily be proved by a calculation with 2×2 matrices (since everything takes place is the two-dimensional subspace spanned by ψ and ϕ, expect when ϕ = zψ, z ∈ T, in which case (5.75) reads 0 = 0 and hence is true also). Since K1 is linear as well as isometric with respect to the trace-norm, we have

K1(eψ ) − K1(eϕ )1 = K1(eψ − eϕ )1 = eψ − eϕ 1,

and hence, by (5.75), Tr(K1(eψ )K1(eϕ )) = Tr (eψ eϕ ). Eq. (5.70) then gives (5.8). 5.3 Equivalence between the six symmetry theorems 139

We move on to (5.71). The main concern is that this expression be well defined, since in case some eigenvalue λ > 0ofρ is degenerate (necessarily with finite mul- tiplicity, even in infinite dimension, since ρ is compact), the basis of the eigenspace ∑ λ Hλ that takes part in the sum i ieυi is far from unique. This is settled as follows:

Lemma 5.16. Let W : P1(H) → P1(H) be a bijection that satisﬁes (5.8), let L ⊂ H (υ ) (υ ) be a (ﬁnite-dimensional) subspace, and let j and i be bases of L. Then

∑W(eυ )=∑W(eυ ). (5.76) j i j i

Proof. As usual, for projections e and f on H we write e ≤ f iff eH ⊆ fH. From 2 (B.212) and (B.214) we have ∑ j |υ j,ψ| ≤ 1 for any unit vector ψ ∈ H, with ψ ∈ ≤ ∑ ( )= equality iff L. In other words, eψ eL iff j Tr eυ j eψ 1. Furthermore, by W( ) ∑ W( ) (5.8) the images eυ j remain orthogonal; hence j eυ j is a projection, and e ≤ ∑ j W(eυ ) iff ∑ j Tr (W(eυ )e)=1. By (5.8), this condition is satisfied for e = j j W( υ ) W( ) ≤ ∑ W( υ ) W( ) e i , so that eυ j e j for each j. Since also the projections eυi i ∑ W( ) ≤ ∑ W( υ ) are orthogonal, this gives i eυi j e j . Interchanging the roles of the two bases gives the converse, yielding (5.76). Finally, to prove bijectivity of the correspondence K ↔ W, we need the property K λ = λ K( ), ∑ ieυi ∑ i eυi (5.77) i i since this implies that K is determined by its action on P1(H) ⊂ D(H). In finite dimension this follows from convexity of K, and we are done. In infinite dimension, K ∑ λ we in addition need continuity of , as well as convergence of the sum i ieυi not only in the operator norm (as follows from the spectral theorem for self-adjoint compact operators), but also in the trace norm: for finite n,m,

m m m λ ≤ |λ | = λ , ∑ ieυi 1 ∑ i eυi 1 ∑ i i=n i=n i=n = ∑ λ = , → ∞ since eυi 1 1. Because i i 1, the above expression vanishes as n m , ρ = ∑n λ ( ) whence n i=1 ieυi is a Cauchy sequence in B1 H , which by completeness of the latter converges (to an element of D(H), as one easily verifies). The proof of continuity is completed by noting that K is continuous with respect to the trace norm, for it is isometric and hence bounded (see step 2 above). K It is enlightening to give a rather more conceptual proof that |P1(H) satisfies (5.8), which is based on a result to be used more often in the future. In what follows, for any convex set C, the notation Ab(K) stands for the real vector space of bounded affine functions f : C → R, that is, bounded functions satisfying

f (tx+(1 −t)y)=tf(x)+(1 −t) f (y), x,y ∈ C,t ∈ (0,1). (5.78)

It is easily checked that Ab(K) with the supremum-norm is a real Banach space. 140 5 Symmetry in quantum mechanics

Proposition 5.17. For any Hilbert space H we have an isometric isomorphism ∼ Ab(D(H)) = B(H)sa, (5.79) f ↔ a; (5.80) f (ρ)=Tr (ρa), (5.81) which preserves the unit (i.e., 1D(H) ↔ 1H ) as well as the order (i.e, f ≥ 0 iff a ≥ 0). ∼ Note that under the identiﬁcation D(H) = Sn(B(H)) (where in ﬁnite dimension the normal state space Sn(B(H)) simply coincides with the state space S(B(H))), where ρ ↔ ω as in (2.33), i.e., ω(a)=Tr (ρa), the above isomorphism simply reads ∼ Ab(Sn(B(H))) = B(H)sa, (5.82) aˆ ↔ a; (5.83) aˆ(ω)=ω(a). (5.84)

Proof. It is clear that for each a ∈ B(H)sa the function f : ρ → Tr (ρa) (or, equivalently,a ˆ : ω → ω(a)) is afﬁne as well as real-valued, and is bounded by (A.100) (supplemented, if dim(H)=∞, by Lemma B.142), noting that ρ1 = 1 for ρ ∈ D(H), and in fact (B.483) yields the equality f ∞ = a (or aˆ∞ = a). Conversely, f ∈ Ab(D(H)) deﬁnes a function Q : H → R by

Q(0)=0; (5.85) 2 Q(ψ)=ψ f (eψ/ψ)(ψ = 0). (5.86)

This function is clearly bounded on the unit ball of H,asin

2 |Q(ψ)|≤f ∞ψ . (5.87)

To check that Q in fact defines a quadratic form on H, we verify the properties (A.8) - (A.9). The first is trivial. The second follows from the easily verified identity

te v+w +(1 −t)e v−w = se v +(1 − s)e w , (5.88) v+w v−w v w where v,w = 0, v = w, and the coefﬁcients s,t are given by

v + w2 t = ; (5.89) 2(v2 + w2) v2 s = . (5.90) v2 + w2

The afﬁne property (5.78) then immediately yields (A.9). According to Proposition B.79, we obtain a unique operator a ∈ B(H)sa such that Q(ψ)=ψ,aψ, i.e.,

ψ,aψ = f (eψ ), ψ ∈ H,ψ = 1. (5.91) 5.3 Equivalence between the six symmetry theorems 141

Since also ψ,aψ = Tr (eψ a), we have established (5.81) for each ρ = eψ , where ψ ∈ ,ψ = ρ = ∑ λ H 1. To extend this result to general density operators i ieυi ,we use (A.100) as well as convergence of the above sum in the trace norm ·1, cf. the proof of Lemma 5.16; the details are analogous to the proof of Theorem B.146.

Proposition 5.18. For any unit vectors ψ,ϕ ∈ H we have

Tr (eψ eϕ )=inf{ f (eψ ) | f ∈ Ab(D(H)),0 ≤ f ≤ 1, f (eϕ )=1}. (5.92)

The virtue of this formula is that the expression on the left-hand side, which deﬁnes the transition probabilities on ∂eD(H)=P1(H), is intrinsically given by the convex structure of D(H). Consequently, any afﬁne bijection of this convex set (which already preserves the boundary) must preserve these probabilities. Proof. By the previous proposition, eq. (5.92) is equivalent to

Tr (eψ eϕ )=inf{ψ,aψ|a ∈ B(H)sa,0 ≤ a ≤ 1,ϕ,aϕ = 1}. (5.93)

Since Tr(eψ eϕ )=ψ,eϕ ψ, we are ready if we can show that the infimum is reached at a = eϕ . Therefore, we prove that for any a as specified we must have ψ,aψ≥Tr (eψ eϕ )=|ϕ,ψ|2. To do so, we are going to find a contradiction if

ψ,aψ < Tr (eψ eϕ ), (5.94) for some such a. Indeed, ϕ,aϕ = 1 with a≤1 (which follows from 0 ≤ a ≤ 1) and ϕ = 1 imply, by Cauchy–Schwarz, that aϕ = ϕ. Since a∗ = a (by positivity of ⊥ ⊥ a), we also have a : (C·ϕ) → (C·ϕ) , so we may write a = eϕ +a , with a ϕ = 0 and a mapping (C · ϕ)⊥ to itself. Then a ≥ 0 implies a ≥ 0. If (5.94) holds, then ψ,a ψ < 0, which contradicts positivity of a (and hence of a). We now turn to the equivalence between Jordan’s Theorem and Kadison’s Theorem. Proposition 5.19. There is an isomorphism of groups between: • The group of afﬁne bijections K : D(H) → D(H); • The group of Jordan automorphisms J : B(H)sa → B(H)sa, such that for any a ∈ B(H)sa one has

Tr (K(ρ)a)=Tr (ρJ(a)) (ρ ∈ D(H)). (5.95)

This immediately follows from the following lemma (of independent interest): Lemma 5.20. 1. There is a bijective correspondence between: • afﬁne bijections K : D(H) → D(H); • unital positive (i.e. order-preserving) linear bijections α : B(H)sa → B(H)sa,

such that for any a ∈ B(H)sa one has (5.95). 2. A map α : B(H) → B(H) is a unital positive linear bijection iff it is a Jordan automorphism. 142 5 Symmetry in quantum mechanics

Proof. 1. An afﬁne bijection K : D(H) → D(H) induces an isomorphism

∗ K : Ab(D(H)) → Ab(D(H)); (5.96) f → f ◦ K, (5.97)

which is evidently unital, positive, and isometric. Consequently, by Proposition ∗ 5.17, K corresponds to some isomorphism α : B(H)sa → B(H)sa, which necessarily shares the properties of being unital, positive, and isometric; this follows abstractly from the proposition, but may also be verified directly from (5.95). Conversely, such a map α yields a map K directly by (5.95); to see this, we identify D(H) with the normal state space of B(H) through ρ ↔ ω, as usual, cf. (2.33), and note that Kω is the state defined by (Kω)(a)=ω(α(a)), or briefly Kω = ω ◦ α. This is often written as K = α∗, and for future reference we write

α∗ω(a)=ω(α(a)). (5.98)

2. The nontrivial direction of the proof (i.e. positive etc. ⇒ Jordan) is based on a number of facts from operator theory:

a. Unital positive linear maps maps on B(H)sa preserve P(H), cf. (2.164). b. Any two projections e and f are orthogonal (ef = 0) iff e + f ≤ 1H (easy). c. Any a ∈ B(H)sa is a norm-limit of ﬁnite sums of the kind ∑i λiei, where λi ∈ R and the ei are mutually orthogonal projections (this follows from the spectral theorem for bounded self-adjoint operators in the form of Theorem B.104) d. Any unital positive linear map α : B(H)sa → B(H)sa is continuous. Since

−a·1H ≤ a ≤−a·1H (a ∈ B(H)sa), (5.99)

by (C.83), applying the positive map α and using α(1H )=1H yields

−a·1H ≤ α(a) ≤−a·1H .

This is possible only if α(a)≤a, and hence α is continuous with norm bounded by α≤1. In fact, since a is unital we have α = 1. Therefore, any unital positive linear map α preserves orthogonality of projections, so if a = ∑i λiei (ﬁnite sum), then α( 2)=α λ 2 = λ 2α( )= λ λ α( )α( )=α( )2, a ∑ i ei ∑ i ei ∑ i j ei e j a (5.100) i i i, j

since eie j = δije j and by the above comment also α(ei)α(e j)=δijα(e j).By continuity of α, this property extends to arbitrary a ∈ B(H)sa. Finally, since

◦ = 1 (( + )2 − 2 − 2), a b 2 a b a b (5.101) preserving squares as in (5.100) implies preserving the Jordan product ◦. 5.3 Equivalence between the six symmetry theorems 143

We now turn to the equivalence between Ludwig symmetries and Jordan ones. Proposition 5.21. There is an isomorphism of groups between: • The group of afﬁne order isomorphism L : E (H) → E (H); • The group of Jordan automorphisms J : B(H)sa → B(H)sa.

Proof. Since L is an order isomorphism, it satisfies L(0)=0 (as well as L(1H )= 1H ), since 0 is the bottom element of E (H) as a poset (and 1H is its the top element). As in the proof of Lemma 5.11, one shows that this property plus convexity implies L(ta)=tL(a) and L(a + b)=L(a)+L(b) whenever defined. Defining J by

J(0)=0; (5.102) J(a)=s · L(a/s)(a > 0,s ≥a); (5.103) J(a)=−J(−a)(a < 0), (5.104) where a > 0 means a ≥ 0 and a = 0, and a < 0 means −a ≥ 0 and a = 0, once again the reasoning near the end of the proof of Lemma 5.11 shows that J is linear; it is a untital order-preserving bijection by construction. Hence J is a Jordan automorphism by Lemma 5.20.2 Of course, instead of (5.104) one could equivalently have deﬁned J on general a ∈ B(H)sa by J(a)=J(a+) − J(a−), using the (by now hopefully familiar) decomposition a = a+ − a− with a± ≥ 0 and a+a− = 0. Conversely, once again using Lemma 5.20.2, a Jordan automorphisms (5.11) preserves order as well as the unit, so that the inequality 0 ≤ a ≤ 1H characterizing a ∈ E (H) is preserved, i.e., 0 ≤ J(a) ≤ 1H . Thus J preserves E (H), where it preserves order. Convexity is obvious, since L = J|E (H) comes from a linear map. The equivalence between Jordan’s Theorem and von Neumann’s Theorem (provided dim(H) ≥ 3) hinges on the following corollary of Gleason’s Theorem (cf. §D.1). Corollary 5.22. Let dim(H) > 2. Then an isomorphism N of P(H) as an orthocomplemented lattice has a unique extension to a linear map α : B(H)sa → B(H)sa, which is (automatically) invertible, unital, and positive. N P( ) Proof. According to Lemma D.2, preserves all suprema in H . Since we have ∑i ei = ei for any family of mutually orthogonal projections and since N by deﬁ- nition preserves the orthocomplementation e⊥ = 1 − e and hence preserves orthogonality of projections, we may compute N ∑ei = N ei = N(ei)=∑N(ei). (5.105) i i i i

Consequently, for any normal state ω on B(H), the map e → ω ◦N(e) is a probability measure on P(H), which by Gleason’s Theorem has a unique linear extension to B(H) and hence a fortiori to B(H)sa. We use this in order to deﬁne α, as follows. First, let a ∈ B(H)sa and suppose a = ∑ j λ j f j for some ﬁnite family ( f j) of projections (not necessarily orthogonal), and some λ j ∈ R. Then ∑ j λ jN( f j) is independent of the particular decomposition of a that has been chosen, so we may put 144 5 Symmetry in quantum mechanics

α(a)=∑λ jN( f j). (5.106) j

= ∑ λ α ( )=∑ λ N( ) α ( ) = To see this, put a j j f j and hence a j j f j , and suppose a α(a). By (B.477) there exists a normal state ω such that ω(α (a)) = ω(α(a)); indeed, each element of B1(H) is a linear combination of at most four density operators, so that each normal linear functional on B(H) is a linear combination of at most four normal states. But since ω ◦N is linear, this implies ω ◦N(a) = ω ◦N(a), which is a contradiction. Hence α (a)=α(a) and accordingly, (5.106) is well de- ﬁned. Because it is independent of the decomposition of a into projections, α is

= ∑ λ = ∑ λ + = ∑ λ + ∑ λ linear: if a j j f j and a j j f j , then a a j j f j j j f j , so that N( + )=N λ + λ = λ N( )+ λ N( )=N( )+N( ). a a ∑ j f j ∑ j f j ∑ j f j ∑ j f j a a j j j j

Similarly, for any t ∈ R we have

N(ta)=N ∑tλ j f j = ∑tλ jN( f j)=t ∑λ jN( f j)=tN(a). j j j

We may now extend α to all of B(H)sa by continuity. Indeed, according to the spectral theorem in the form (B.326), the set of all operators of the form a = ∑ j λ j f j with all f j mutually orthogonal (so that a is given by its spectral resolution) is norm- ( ) = |λ | dense in B H sa. Applying (5.106), and noting that a sup j j , we may estimate

α(a) = ∑λ jN( f j)≤sup{|λ j|}∑N( f j)≤a, j j j since the N( f j) are mutually orthogonal and hence sum to some projection, which has norm 1 (unless a = 0). For general a ∈ B(H)sa, we may therefore define N by N(a)=limn N(an), where each an is of the above (spectral) form and an −a→0. To prove that α is positive, we show that α(a) ≥ 0 whenever a ≥ 0. As in the preceding step, initially suppose that a = ∑ j λ j f j has a finite spectral resolution. Then a ≥ 0iffλ j ≥ 0 for each j, and hence α(a) ≥ 0 by (5.106), since by orthogonality of the N( f j) this equation states the spectral resolution of α(a).Nowifan ≥ 0 and an → a (in norm), then ψ,anψ→ψ,aψ, which must remain positive, so that a ≥ 0. Hence positivity of α on all of B(H)sa follows by continuity. Finally, α inherits invertibility from N, and it is unital by (5.105), taking ei = |υiυi| for some basis (υi) of H (or using the fact that it preserves # = 1H ). Subsequently, we use Lemma 5.20 to further extend α by complex linearity to a Jordan isomorphism of B(H); see Definition 5.1. Finally, the equivalence between weak Jordan symmetries and Bohr symmetries follows from Hamhalter’s Theorem 9.4, whereas Theorem 9.7 strengthens this to an equivalence between Jordan symmetries and Bohr symmetries. The proof of these theorems does not seem to simplify in the special case at hand, i.e. A = B(H). 5.4 Proof of Jordan’s Theorem 145 5.4 Proof of Jordan’s Theorem

In view of the equivalence between the six parts of Theorem 5.4, we only need to prove one of them. In the literature, one only ﬁnds proofs of Jordan’s Theorem and of Wigner’s Theorem, and we present each of these (surprisingly but instructively, these proofs look completely different). We start with Jordan’s Theorem:

Theorem 5.23. Any Jordan automorphism JC of B(H) is given by either

∗ JC(a)=αu(a) ≡ uau , (5.107) where u is unitary (and is determined by JC up to a phase), or by J ( )=α ( ) ≡ ∗ ∗, C a u a ua u (5.108) where u is anti-unitary (and is determined by JC up to a phase, too). The difficult part of the proof is Theorem C.175, which implies: Proposition 5.24. A Jordan automorphism α of B(H) is either an automorphism or an anti-automorphism. Recall that an automorphism of B(H) is a linear bijection α : B(H) → B(H) that satisfies α(a∗)=α(a)∗ and α(ab)=α(a)α(b);ananti-automorphism, on the other hand, satisfies the first property whilst the latter is replaced by α(ab)=α(b)α(a). Clearly, both automorphisms and anti-automorphisms are Jordan automorphisms. Granting this result, we may deal with the two cases separately.

Proposition 5.25. Any automorphism α : B(H) → B(H) takes the form α = αu, see (5.107), where u : H → H is unitary, uniquely determined by α up to a phase. The proof uses the following lemmas. The ﬁrst follows from Theorem C.62.4. Lemma 5.26. If α : B(H) → B(H) is an automorphism and a ∈ B(H), then

α(a) = a. (5.109)

Lemma 5.27. If α : B(H) → B(H) is an automorphism and e ∈ B(H) is a one- dimensional projection, then so is α(e). Proof. It should be obvious that automorphisms α preserve projections e (whose defining properties are e2 = e∗ = e). Furthermore, α preserves order, i.e., if a ≥ 0 (in that, as always, ψ,aψ≥0 for each ψ ∈ H, or, equivalently, a = b∗b), then α(a) ≥ 0 (this is clear from the second way of expressing positivity). Consequently, if a ≤ b (in that b − a ≥ 0), then α(a) ≤ α(b). We notice that if we define e ≤ f iff eH ⊆ fH, then e ≤ f iff e ≤ f as self-adjoint operators (in that ψ,eψ≤ψ, f ψ for each ψ ∈ H); see Proposition C.170. With respect to the ordering ≤ the one- dimensional projections e are atomic, in the sense that 0 ≤ e (but e = 0) and if 0 ≤ f ≤ e, then either f = 0or f = e. Now automorphisms of the projection lattice B(H) restrict to isomorphisms of P(H), which preserve atoms (as these are intrinsically defined by the partial order). 146 5 Symmetry in quantum mechanics

We are now ready for the (constructive!) proof of Proposition 5.25. Proof. For some ﬁxed unit vector χ ∈ H, take the corresponding one-dimensional projection eχ and deﬁne a new unit vector ϕ (up to a phase) by

−1 eϕ = α (eχ ). (5.110)

Now any ψ ∈ H may be written as ψ = aϕ, for some a ∈ B(H). Attempt to deﬁne an operator u by uψ = α(a)χ, i.e.,

uaϕ = α(a)χ. (5.111)

This looks dangerously ill-deﬁned, since many different operators a may give rise to the same ψ. Fortunately, we may compute

aϕH = aeϕ ϕH = aeϕ B(H) = α(aeϕ )B(H)

= α(a)α(eϕ )B(H) = α(a)eχ B(H) = α(a)χH

= uaϕH , so that if aϕ = bϕ, then α(a)χ = α(b)χ and hence u is well deﬁned. By this computation u is also isometric and since it is clearly surjective, it is unitary. The property α(a)=uau∗ is equivalent to ua = α(a)u, which in turn is equivalent to uabϕ = α(a)ubϕ for any b ∈ B(H), which by deﬁnition of u is the same as

α(ab)χ = α(a)α(b)χ. (5.112)

But this holds by virtue of α being an automorphism. Finally, all arbitrariness in u lies in the lack of uniqueness of ϕ given its deﬁnition (5.110).

Proposition 5.28. Any antiautomorphism α : B(H) → B(H) takes the form α = αu, cf. (5.108), where u : H → H is anti-unitary, uniquely determined by α up to a phase.

Proof. Pick an arbitrary anti-unitary operator J : H → H and deﬁne

β : B(H) → B(H); β(a)=Ja∗J∗. (5.113)

Then α ◦ β is an automorphism, to which Proposition 5.25 applies, so that

α ◦ β(a)=ua˜ u˜∗, (5.114) for some unitaryu ˜. Hence

α(a)=α(β ◦ β −1(a)) = α ◦ β(J∗a∗J)=uJ˜ ∗a∗Ju˜∗, so that α(a)=ua∗u∗ with u = uJ˜ ∗. The precise lack of uniqueness of u is inherited from the unitary case. 5.5 Proof of Wigner’s Theorem 147 5.5 Proof of Wigner’s Theorem

We recall Wigner’s Theorem, i.e. Theorem 5.4.1:

Theorem 5.29. Each bijection W : P1(H) → P1(H) that satisﬁes

Tr (W(e)W( f )) = Tr (ef), (e, f ∈ P1(H)), (5.115)

∗ is given by W(e)=ueu ≡ αu(e), where the operator u is either unitary or anti- unitary, and is uniquely determined by W up to a phase.

The problem is to lift a given map W : P1(H) → P1(H) that satisﬁes (5.115) to either a unitary or an anti-unitary map u : H → H such that

∗ W(eψ )=euψ = ueψ u . (5.116)

Suppose W(eψ )=eψ . Since ezψ = eψ for any z ∈ T, and likewise for eψ , this means that uψ = zψ for some z ∈ T; the problem is to choose the z’s coherently all over the unit sphere of H. There are many proofs in the literature, of which the following one—partly based on an earlier proof by Bargmann (1964)—has the advantage of making at least the construction of u explicit (at the cost of opaque proofs of some crucial lemma’s). We assume dim(H) > 2, since H = C2 has already been covered.

Fix unit vectors ψ ∈ H and ψ ∈ W(eψ )H; clearly, ψ is unique up to multiplication by z ∈ T, whose choice turns out to completely determine u (i.e., the ambiguity in ψ is the only one in the entire construction). For a modest start, we put

uψ = ψ . (5.117)

Lemma 5.30. If V ⊂ H is a k-dimensional subspace (where k < ∞), then there is a unique k-dimensional linear subspace V ⊂ H with the following property:

For all unit vectors ψ ∈ H, we have ψ ∈ ViffW(eψ )H ⊂ V . (υ ,...,υ ) υ ∈ υ ∈ Proof. Pick a basis 1 k of V and ﬁnd unit vectors i H such that i W( ) = ,..., eυi H, i 1 k. Then, using (5.115) we compute

2 2 |υ ,υ | = Tr (eυ eυ )=Tr (W(eυ )W(eυ )) = Tr (eυ eυ )=|υi,υ j| = δij, i j i j i j i j (υ ,...,υ ) so that the vectors 1 k form an orthonormal set and hence form a basis of their linear span V . Now, as mentioned below (B.214), we have ψ ∈ V iff k |υ ,ψ|2 = ψ ∈ k |υ ,ψ |2 = W ∑i=1 i 1 and similarly V iff ∑i=1 i 1. Since preserves transition probabilities, a computation similar to one just given yields

k k |υ ,ψ|2 = |υ ,ψ |2, ∑ i ∑ i (5.118) i=1 i=1 so that both sides do or do not equal unity, and hence ψ ∈ V iff ψ ∈ V . 148 5 Symmetry in quantum mechanics

Wigner’s Theorem for H = C2 (i.e. Theorem 5.7) implies: Lemma 5.31. If V and V are related as in Lemma 5.30, and

dim(V)=dim(V )=2, (5.119)

then there is a unitary or anti-unitary operator uV : V → V such that W( )= ∗ , e uV euV (5.120) for any one-dimensional projection e ∈ P1(V), where P1(V) ⊂ P1(H) consists of all e ∈ P1(H) with eH ⊂ V. Moreover, uV is unique up to a phase.

=∼ Proof. A choice of basis for both V and V gives unitary isomorphisms u : V → C2 =∼ and u : V → C2, which jointly induce a map

−1 2 2 W ≡ u Wu : P1(C ) → P1(C ). (5.121)

This maps satisﬁes the hypotheses of Wigner’s Theorem in d = 2, and so it is (anti-) 2 2 unitarily induced as W = αv, where v : C → C is (anti-) unitary. Then the operator −1 uV =(u ) vu does the job; its lack of uniqueness stems entirely from v.

Lemma 5.32. Given a Wigner symmetry W, the ensuing operator uV is either unitary or anti-unitary for all two-dimensional subspaces V ⊂ H (simultaneously).

Proof. We ﬁrst design a “unitarity test” for W. Deﬁne a function

T : P1(H) × P1(H) × P1(H) → C; (5.122) T(e, f ,g)=Tr (efg), (5.123) ( , , )=ψ ,ψ ψ ,ψ ψ ,ψ . T eψ1 eψ2 eψ3 1 2 2 3 3 1 (5.124)

Let V ⊂ H be two-dimensional and pick an orthonormal basis (υ1,υ2). Deﬁne √ √ χ1 = υ1, χ2 =(υ1 − υ2)/ 2, χ3 =(υ1 − iυ2)/ 2. (5.125)

A simple computation then shows that

( , , )= 1 ( + ). T eχ1 eχ2 eχ3 4 1 i (5.126) It follows from (5.124) that for u unitary and v anti-unitary, we have ( , , )= ( , , ) T euψ1 euψ2 euψ3 T eψ1 eψ2 eψ3 ; (5.127) ( , , )= ( , , ). T evψ1 evψ2 evψ3 T eψ1 eψ2 eψ3 (5.128) Eq. (5.126) implies that if W : V → V is (anti-) unitarily implemented, we have

(W( ),W( ),W( )) = ( , , )= 1 ( ± ), T eχ1 eχ2 eχ3 T euχ1 euχ2 euχ3 4 1 i (5.129) 5.5 Proof of Wigner’s Theorem 149 with a plus sign if u is unitary and a minus sign if u is anti-unitary. Now take a second pair (V˜ ,V˜ ) as above, and pick a basis (υ˜1,υ˜2) of V˜ , with associated vectors

(χ˜1, χ˜2, χ˜3), as in (5.125). Suppose u : V → V implementing W is unitary, whereas u˜ : V˜ → V˜ implementing W is anti-unitary. It then follows from (5.129) that

(W( ),W( ),W( )) = ( , , )= 1 ( + ) T eχ1 eχ2 eχ3 T euχ1 euχ2 euχ3 4 1 i ; (5.130) (W( ),W( ),W( )) = ( , , )= 1 ( − ). T eχ˜1 eχ˜2 eχ˜3 T eu˜χ˜1 eu˜χ˜2 eu˜χ˜3 4 1 i (5.131)

In view of (C.637), the following expression deﬁes a metric d on P1(H): 2 d(eψ ,eϕ )=ωψ − ωϕ = eψ − eϕ 1 = 2 1 −|ϕ,ψ| , (5.132) with respect to which both W and T are continuous (the latter with respect to the 3 product metric on P1(H) , of course). Let t → (υ1(t),υ2(t)) be a continuous path of orthonormal vectors (i.e., in H ×H), with associated vectors (χ1(t), χ2(t), χ3(t)),as in (5.125). Then the function f (t)=T(W(χ1(t)),W(χ2(t)),W(χ3(t))) is continu- 1 ( ± ) ( ) ous, and by (5.129) it can only take the values 4 1 i . Hence f t must be constant. However, taking a path such that (υ1(0),υ2(0))=(υ1,υ2) and (υ1(1),υ2(1)) = (υ ,υ ) ( )= 1 ( + ) ( )= 1 ( − ) ˜1 ˜2 ,gives f 0 4 1 i and f 1 4 1 i , which is a contradiction. Lemma 5.33. Wigner’s Theorem holds for three-dimensional Hilbert spaces.

3 Proof. Let (υ1,υ2,υ3) be some basis of of H (like the usual basis of H = C ). We ﬁrst show that if W is the identity if restricted to both span(υ1,υ2) and span(υ1,υ3), then W is the identity on H altogether. To this end, take ψ = ∑i ciυi, initially with ∈ R\{ } ψ ∈ W( ) ψ = υ c1 0 . Take a unit vector eψ , with ∑i ci i. By the ﬁrst assumption on W we have |υ,ψ | = |υ,ψ| for any unit vector υ ∈ span(υ1,υ2). Taking √ √ υ = υ1, υ = υ2, υ =(υ1 + υ2)/ 2, υ =(υ1 + iυ2)/ 2, (5.133) gives the equations | | = | |, | | = | |, | + | = | + |, | − | = | − |, c1 c1 c2 c2 c1 c2 c1 c2 c1 ic2 c1 ic2 (5.134) = respectively. By a choice of phase we may and will assume c1 c1, in which case = the only solution is c2 c2 (geometrically, the solution c2 lies in the intersection of three different circles in the complex plane, which is either empty or consists W = of a single point). Similarly, the second assumption on gives c3 c3, whence

ψ = ψ. The case c1 = 0 may be settled by a straightforward limit argument, since inner products (and hence their absolute values) are continuous on H × H. Given a Wigner symmetry W : P1(H) → P1(H), we now construct u as follows.

1. Fix a basis (υ1,υ2,υ3) with “image” (υ ,υ ,υ ) under W, i.e, W(eυ )=eυ . 1 2 3 i i 2. The unitarity test in the proof of Lemma 5.32 settles if the operators should be chosen to be unitary or anti-unitary; for simplicity we assume the unitary case. → υ = υ = , , 3. Deﬁne a unitary u1 : H H by u1 i i for i 1 2 3, and subsequently de- W = α ◦ W ﬁne 1 u1 , which (being the composition of two Wigner symmetries) 150 5 Symmetry in quantum mechanics

W ( )= = , , W is a Wigner symmetry. Clearly, 1 eυi eυi (i 1 2 3), so that 1 maps P1(H(12)) to itself, where H(12) ≡ span(υ1,υ2). Hence Lemma 5.31 gives a uni- → W α tary mapu ˜1 : H(12) H(12) such that the restriction of 1 to H(12) is u˜1 . → = −1 υ = υ 4. Define a unitary u2 : H H by u2 u˜1 on H(12) and u2 3 3, followed by the W = α ◦W W ( )= = , , Wigner symmetry 2 u2 1. By construction, 2 eυi eυi for i 1 2 3) (W2 is even the identity on P1(H(12))), so that W2 maps P1(H(13)) to itself, where H(13) ≡ span(υ1,υ3). Hence the restriction of W2 to H(13) is implemented by a unitaryu ˜2 : H(13) → H(13), whose phase may be fixed by requiringu ˜2υ1 = υ1. → = −1 υ = υ 5. Similarly to u2, we define u3 : H H by u3 u˜2 on H(13) and u3 2 2,so that u3 is the identity on H(12). Of course, we now define a Wigner symmetry W = α ◦ W = α ◦ α ◦ α ◦ W, 3 u3 2 u3 u2 u1 (5.135)

which by construction is the identity on both P1(H(12)) and P1(H(13)), and so by the ﬁrst part of the proof it must be the identity on all of P1(H). Hence

−1 −1 −1 W = α −1 ◦ α −1 ◦ α −1 = αu (u = u u u ). u1 u2 u3 1 2 3 Lemma 5.34. As in Lemma 5.30, if dim(V)=dim(V )=3, then there is a unitary → W( )= ∗ ∈ P ( ) or anti-unitary operator uV : V V such that e uV euV for any e 1 V , Proof. Given Lemma 5.33, the proof is practically the same as for Lemma 5.31.

We now finish the proof of Wigner’s Theorem. We assume that the outcome of Lemma 5.32 is that each uV is unitary; the anti-unitary case requires obvious modifications of the argument below. The first step is, of course, to define u(λψ)= λuψ, λ ∈ C (so this would have been λuψ in the anti-unitary case). Let ϕ ∈ H be linearly independent of ψ and consider the two-dimensional space V spanned by ψ and ϕ. Define u(ϕ)=uV ϕ. With (5.117), this defines u on all of H. To prove that u is linear, take ϕ1 and ϕ2 linearly independent of each other and of ψ, so that the linear span V3 of ψ, ϕ1, and ϕ2 is three-dimensional. Let Vi be the two-dimensional ψ ϕ = , ϕ = ϕ linear span of and i, i 1 2. Then u i uVi i, where the phase of uVi is fixed → W by (5.117). Let w : V3 V3 be the unitary that implements according to Lemma 5.33.2, with phase determined by (5.117). Since uV1 and uV2 and w are unique up to a phase and this phase has been fixed for each in the same way, we must have = = ψ ϕ + ϕ uV1 w|V1 and uV2 w|V2 . Finally, we have V12 spanned by and 1 2, and by = the same token, uV12 w|V12 .Noww is unitary and hence linear, so (ϕ + ϕ )= (ϕ + ϕ )= (ϕ + ϕ )= (ϕ )+ (ϕ ) u 1 2 uV12 1 2 w 1 2 w 1 w 2 = (ϕ )+ (ϕ )= (ϕ )+ (ϕ ), uV1 1 uV2 2 u 1 u 2 since this is how u was defined. Since each uV is unitary, so is u, and similarly it is easy to verify that u implements W, because each uV does so. 5.6 Some abstract representation theory 151 5.6 Some abstract representation theory

Since all symmetries we have considered (named after Wigner, Kadison, Jordan, Ludwig, von Neumann, and Bohr) are implemented by either unitary or anti-unitary operators, which are determined (by the given symmetry) only up to a phase z ∈ T, the quantum-mechanical symmetry group G H of a Hilbert space H is given by

H G =(U(H) ∪Ua(H))/T, (5.136) where U(H) is the group of unitary operators on H, and Ua(H) is the set of anti- unitary operators on H; the latter is not a group (since the product of two anti- unitaries is unitary) but their union is. Furthermore, T is identiﬁed with the normal subgroup T ≡ T·1H = {z·1H | z ∈ T} of U(H)∪Ua(H) (and also of U(H)) consist- ing of multiples of the unit operators by a phase; thus the quotient G H is a group. The fact that G H rather than U(H) is the symmetry group of quantum mechanics has profound consequences (one of which is our very existence), which we will study from §5.10 onwards. However, this material relies on the theory of “ordinary” (i.e., non-projective) unitary representations, which we therefore review ﬁrst. Namely, let G be a group. In mathematics, the natural kind of action of G on a Hilbert space H is a unitary representation, i.e., a homomorphism

u : G → U(H), (5.137)

−1 −1 ∗ so that u(x) = u(x )=u(x) and u(x)u(y)=u(xy), which imply u(e)=1H . As to the possible continuity properties of unitary representations in case that G is a topological group (i.e., a group G that is also a topological space, such that group multiplication G × G → G and inverse G → G are continuous), one should equip U(H) with the strong operator topology (as opposed to the norm topology). Proposition 5.35. If u : x → u(x) is a unitary representation of some locally compact group G on a Hilbert space H, then the following conditions are equivalent: 1. The map G × H → H, (x,ψ) → u(x)ψ, is continuous; 2. The map G → U(H),x→ u(x), is continuous in the strong topology on U(H).

Proof. Strong continuity means that if xλ → x in G, then for each ψ ∈ H we have (u(xλ )−u(x))ψ→0. This is clearly implied by the ﬁrst kind of continuity, giving 1 ⇒ 2, so let us prove the nontrivial converse. Suppose xλ → x and ψμ → ψ; since G is locally compact, x has a compact neighborhood K and we may assume that each xλ ∈ K.Ifu is strongly continuous, then for any ϕ ∈ H the set {u(y)ϕ,y ∈ K} is compact in H and hence bounded. The Banach–Steinhaus Theorem B.78 gives boundedness of the corresponding operator norms, that is, {u(y),y ∈ K} < CK for some CK > 0. We now estimate

u(xλ )ψμ − u(x)ψ≤u(xλ )ψμ − u(xλ )ψ + (u(xλ ) − u(x))ψ.

The ﬁrst term vanishes as ψμ → ψ since it is bounded by CKψμ −ψ, whereas the second vanishes as xλ → x by the (assumed) strong continuity of u. 152 5 Symmetry in quantum mechanics

Since the first kind of continuity is the usual one for group actions, this justifies the choice of strong continuity as the natural one for unitary representations (to which a pragmatic point may be added: norm continuity is quite rare for unitary representations on infinite-dimensional Hilbert spaces). Things further simplify under mild restrictions on G and H, which are satisfied in all examples of physical interest. Proposition 5.36. If H is separable and G is second countable locally compact (sclc), then each of the two continuity conditions in Proposition 5.35 is in turn equivalent to weak measurability of u, in that for each ϕ,ψ ∈ H the function

x →ϕ,u(x)ψ from G to C is (Borel) measurable. Proof. This spectacular result is due to von Neumann, who more generally proved that a measurable homomorphism between sclc groups is continuous. This implies the claim: ﬁrst, if H is separable, then the group U(H) is sclc in its weak operator topology, so that if the map G → U(H), x → u(x) is weakly measurable, then it is continuous in the weak topology on U(H). Second, for any Hilbert space, weak (operator) continuity of a unitary representation implies strong continuity (so that, given the trivial converse, weak and strong continuity of unitary group representations are equivalent). We only prove this last claim: for x,y ∈ G, we compute

(u(y) − u(x))ψ = u(x)ψ2 + u(y)ψ2 −u(x)ψ,u(y)ψ−u(y)ψ,u(x)ψ = 2ψ2 −ψ,u(x−1y)ψ−ψ,u(y−1x)ψ,

Weak continuity obviously implies that the function x →ψ,u(x)ψ is continuous at the identity e ∈ G,soify = xλ → x, then (u(xλ ) − u(x))ψ→0. In view of this, it is hardly a restriction for a unitary representation of a locally compact group on a Hilbert space to be continuous in the sense of Proposition 5.35, so we always assume this in what follows. Furthermore, any group we consider is locally compact, so this will be a standing assumption, too. An important consequence of this assumption is the existence of a translation-invariant measure on G. Theorem 5.37. Each locally compact group G has a canonical nonzero (outer regular Borel) measure μ, called Haar measure, which is left-invariant in that

dμ(x)Ly f (x)= dμ(x) f (x), (5.138) G G for each f ∈ Cc(G) and y ∈ G, where the left translation Ly of f by y is deﬁned by

−1 Ly f (x)= f (y x). (5.139)

This measure is unique up to scalar multiplication. Moreover, if G is compact, then: 1. μ is ﬁnite and hence can be normalized to a probability measure, i.e.,

μ(G)=1. (5.140) 5.6 Some abstract representation theory 153

2. μ is also right-invariant in that

dμ(x)Ry f (x)= dμ(x) f (x), (5.141) G G

where the right translation Ry of f by y ∈ G is deﬁned by

Ry f (x)= f (xy). (5.142)

3. μ is invariant under inversion, in that dμ(x) f (x−1)= dμ(x) f (x). (5.143) G G Existence is due to Haar and uniqueness was ﬁrst proved by von Neumann. One often writes dx ≡ dμ(x) for Haar measure. Here are some examples: n • For G = R , Haar measure equals Lebesgue measure μL (up to a constant); eqs. (5.139) and (5.141) state the familiar translation invariance of μL. • For G = T,wehave 1 2π dμ(z) f (z)= dθ f (eiθ ). (5.144) T 2π 0

• For G = GLn(R) with X =(xij),wehave

m −n dμ(X)= ∏ dxij|det(X)| , (5.145) i, j=1

which for G = SLn(R) of course simpliﬁes to dμ(X)=∏i, j dxij. Deﬁnition 5.38. A unitary representation u of a group G on a Hilbert space H is irreducible if the only closed subspaces K of H that are stable under u(G) (in the sense that if ψ ∈ K, then u(x)ψ ∈ K for all x ∈ G) are either K = HorK= {0}.

We will often need two important results about irreducibility. The ﬁrst is Schur’s Lemma, in which the commutant S of some subset S ⊂ B(H) is deﬁned by

S = {a ∈ B(H) | ab = ba∀b ∈ S}. (5.146)

Lemma 5.39. A unitary representation u of a group G is irreducible iff

u(G) = C · 1, (5.147) i.e., if au(x)=u(x)a for each x ∈ G implies a = λ · 1H for some λ ∈ C. This follows from Theorem C.90, of which the above lemma is a special case: take A = u(G) ≡ (u(G) ) . The second is part of the Peter–Weyl Theorem. Theorem 5.40. Irreducible representations of compact groups are ﬁnite-dimensional. 154 5 Symmetry in quantum mechanics

Proof. We first reduce the situation to the unitary case: if ·,·, is the given inner product on H, we define a new inner product ·,·, by averaging with respect to Haar measure dx ≡ dμ(x), i.e., ψ,ϕ = dxu(x)ψ,u(x)ϕ. (5.148) G Using (5.141), it is easy to verify that this new inner product makes u unitary. So let u : G → u(H) be an irreducible unitary representation. For each unit vector ϕ ∈ H and x ∈ G, we define the following projection and its G-average:

e ( )ϕ = |u(x)ϕu(x)ϕ|, (5.149) u x

Wϕ = dxeu(x)ϕ . (5.150) G The Weyl operator (5.150) is initially deﬁned as a quadratic form by

ψ1,Wϕ ψ2 = dxψ1,eu(x)ϕ ψ2. (5.151) G The integral exists because the integrand is continuous and bounded, deﬁning a bounded quadratic form by the estimate |ψ1,Wϕ ψ2| ≤ ψ1ψ2, where we assumed (5.140) and used eu(x)ϕ = 1, as (5.149) is a nonzero projection. Thus the operator Wϕ may be reconstructed from its matrix elements (5.151), cf. Proposition B.79. It is easy to verify that [Wϕ ,u(y)] = 0 for each y ∈ G, so that Schur’s Lemma 2 yields Wϕ = λϕ · 1H for some λϕ ∈ C. Hence ψ,Wϕ ψ = λϕ ψ , in other words, 2 2 dx|ψ,u(x)ϕ| = λϕ ψ . (5.152) G

If we now interchange ϕ and ψ and use (5.143) we ﬁnd λϕ ψ2 = λψ ϕ2, so that, taking ψ to be a unit vector, too, since ψ and ϕ are arbitrary we obtain λϕ = λψ ≡ λ, where in fact λ > 0, as follows by taking ψ = ϕ in (5.152). Finally, take n or- thornormal vectors (υ1,...,υn) in H, so that also (u(x)υ1,...,u(x)υn) are orthonormal (since u(x) is unitary), upon which Bessel’s inequality (B.212) gives

n 2 2 ∑ |ψ,u(x)υi| ≤ψ . (5.153) i=1

Integrating both sides over G, taking ψ = 1, and using (5.140) gives

n 2 ∑ dx|ψ,u(x)υi| ≤ 1. (5.154) i=1 G

On the other hand, summing (5.152) over i simply yields nλ, whence nλ ≤ 1, for any n ≤ dim(H). Since λ > 0 this forces dim(H) < ∞. 5.7 Representations of Lie groups and Lie algebras 155 5.7 Representations of Lie groups and Lie algebras

We now assume that G is a Lie group;asin§3.3, for our purposes we may restrict ourselves to linear Lie groups, i.e. closed subgroups of GLn(K) for K = R or C. Let u : G → U(H) be a unitary representation of a Lie group G on some Hilbert space H (assumed strongly continuous). If H is ﬁnite-dimensional, the following operation is unproblematic: for A ∈ g (i.e. the Lie algebra of G) we deﬁne an operator

u (A) : H → H; (5.155) d u (A)= u etA . (5.156) dt |t=0 This gives a linear map u : g → B(H), which satisﬁes

[u (A),u (B)] = u ([A,B]); (5.157) u (A)∗ = −u (A). (5.158)

Note that physicists use Planck’s constant h¯ > 0 and like to write

π(A)=ihu¯ (A), (5.159) so that one has the following commutation relations and self-adjointness condition:

[π(A),π(B)] = ih¯π([A,B]); (5.160) π(A)∗ = π(A). (5.161)

If one knows that u : g → B(H) comes from u : G → U(H), one conversely has

A u (A) − i π(A) u(e )=e = e h¯ . (5.162)

More generally, we call a map ρ : g → B(H) (where H =∼ Cn remains ﬁnite- dimensional, so that ρ : g → Mn(C)), a skew-adjoint representation of g on H if

[ρ(A),ρ(B)] = ρ([A,B]); (5.163) ρ(A)∗ = −ρ(A). (5.164)

The property of irreducibility of such a representation ρ : g → B(H) is defined in the same way as for groups, namely that the only linear subspaces of H =∼ Cn that are stable under ρ(g) are {0} and H. Equivalently, by Schur’s Lemma, ρ(g) is irreducible iff the only operators that commute with all π(A) are multiples of the unit operator. If ρ = u for some unitary representation u(G), it is easy to see that u is irreducible iff u is irreducible. In view of this, it is a reasonable strategy to try and construct irreducible unitary representations u(G) by starting, as it were, from u (g). More precisely, if ρ is some (irreducible) skew-adjoint representation of g, we may ask if there is a (necessarily irreducible) unitary representation u(G) such that ρ = u . Writing exp(ρ) for u, one would therefore hope that 156 5 Symmetry in quantum mechanics u eA ≡ eρ eA = eρ(A), (5.165) as in (5.162). Note that if G is connected, then ρ duly defines u(x) for each x ∈ G through (5.165), since by Lie theory every element x of a connected Lie group is a finite product x = exp(A1)···exp(An) of exponentials of elements (A1,...,An) of g. In general, this hope is in vain, since although each operator exp(A) is unitary, the representation property u(x)u(y)=u(xy) may fail for global reasons. For example, ∼ 3 if G = SO(3), then g = R , with basis (J1,J2,J3), as in (3.66). Define an a priori linear map ρ : g → M2(C) by linear extension of

ρ( )=− 1 σ , Jk 2 i k (5.166) where (σ1,σ2,σ3) are the Pauli matrices (5.42), so that physicists would write

π( )= 1 σ , Jk 2 h¯ k (5.167) cf. (5.159). This is easily checked to give a skew-adjoint representation of g, but it does not exponentiate to a unitary representation of SO(3): as already mentioned 3 after Proposition 5.46, if u is a unit vector in R , then a rotation Rθ (u) around the u-axis by an angle θ ∈ [0,2π] is represented by

u(Rθ (u)) = cos(θ/2) · 12 + isin(θ/2)u · σ. (5.168)

2 Consequently, u(Rπ (u)) = iu · σ, so that u(Rπ (u)) = −12, although within SO(3) one has Rπ (u)2 = e, the unit of SO(3), so that u(Rπ (u))2 = u(Rπ (u)2). However, ρ does exponentiate to a representation of SU (2), which happens to be the universal covering group of SO(3). This is typical of the general situation, which we state without proofs. We ﬁrst need a reﬁnement of Lie’s Third Theorem: Theorem 5.41. Let G be a connected Lie group G with Lie algebra g. There exists a simply connected Lie group G,˜ unique up to isomorphism, such that: • The Lie algebra of Gis˜ g. • G =∼ G˜/D, where D is a discrete normal subgroup of the center of G.˜ ∼ • D = π1(G), i.e., the fundamental group of G, which is therefore abelian.

For example, for G = SO(3) we have G˜ = SU (2) and D = Z2, cf. Proposition 5.46.

Theorem 5.42. Let G1 and G2 be Lie groups, with Lie algebras g1 and g2, respectively, and suppose that G1 is simply connected. Then every Lie algebra homomorphism ϕ : g1 → g2 comes from a unique Lie group homomorphism Φ : G1 → G2 through ϕ = Φ , where (realizing G1 and G2 as matrices) d Φ (X)= Φ etX . (5.169) dt |t=0 ∼ Let H be a ﬁnite-dimensional Hilbert space, so that B(H) = Mn(C), where n = ∼ n dim(H), and take U(H) = Un(C) to be the group of all unitary matrices on C . The Lie algebra un(C) of Un(C) consists of all skew-adjoint n × n complex matrices. Since irreducibility is preserved under the correspondence u(G) ↔ u (g), we infer: 5.7 Representations of Lie groups and Lie algebras 157

Corollary 5.43. Let G be a simply connected Lie group with Lie algebra g. Any ﬁnite-dimensional skew-adjoint representation π : g → un(C) of g comes from a unique unitary representation u(G) through (5.156), in which case we have

eu (A) = u eA (A ∈ g). (5.170)

Thus there is a bijective correspondence between finite-dimensional unitary representations of G and finite-dimensional skew-adjoint representations of g. In particular, if G is compact, this specializes to a bijective correspondence between unitary irreducible representations of G and skew-adjoint irreducible representations of g. If G =∼ G˜/D is connected but not simply connected, then a finite-dimensional skew-adjoint representation ρ : g → B(H) exponentiates to a unitary representation u : G → U(H) iff the representation exp(ρ) : G˜ → U(H) is trivial on D. For example, G = SO(3), the last condition is satisfied for the irreducible representations with integer spins j ∈ N (as well as for j = 0), see §5.8. A similar construction is possible when H is infinite-dimensional, except for the fact that the derivative in (5.156) may not exist. For example, G = R has its canonical regular representation on H = L2(R), defined by u(a)ψ(x)=ψ(x−a), in which case (5.159) gives some multiple of the momentum operator −ihd¯ /dx. This operator is unbounded and hence is not defined on all of H, see also §5.11 and §5.12. As in Stone’s Theorem 5.73, this problem is solved by finding a suitable domain in H on which the underlying limit, taken strongly, does exist. This is the Garding˚ domain = ( )ψ, ∈ ∞( ),ψ ∈ , DG u f f Cc G H (5.171)

∈ ∞( ) ∈ 1( ) ( ) where for each f Cc G (or even f L G ) the operator u f is deﬁned by u ( f )= dx f(x)u(x). (5.172) G

Like the derivative u , this integral is most easily deﬁned weakly, i.e., the (bounded) operator u ( f ) is initially deﬁned as a bounded quadratic form Q(ϕ,ψ)= dx f(x)ϕ,u(x)ψ, (5.173) G from which the operator u ( f ) may be reconstructed as in Proposition B.79. Note that the function x →ϕ,u(x)ψ is in Cb(G), so that the integral (5.173) exists.

It can be shown that DG is dense in H, as well as invariant under u (g),inthe sense that if ψ ∈ DG, then u (A)ψ ∈ DG for any A ∈ g. Furthermore, for each ϕ ∈ DG the function x → u(x)ϕ from G to H is smooth (if G is unimodular this property even characterizes DG). The commutation relations (5.157) then hold on DG,but the equalities (5.164) do not: one has to choose between (5.157) and (5.164), since the latter holds for the closure of each π(A) (i.e., each iρ(A) is essentially self- adjoint on DG), whose domain however depends on A: there is no common domain on which each iρ(A) is self-adjoint and the commutation relations (5.157) hold. 158 5 Symmetry in quantum mechanics 5.8 Irreducible representations of SU (2)

One of the most important groups in quantum physics is SU (2), both as an internal symmetry group—e.g. of the Heisenberg model of ferromagnetism, of the weak nu- clear interaction, and possibly also of (loop) quantum gravity—and as a spatial symmetry group in disguise (all projective unitary representations of SO(3) come from unitary representations of SU (2), preserving irreducibility, cf. Corollary 5.61). In this section we review the well-known classification and construction of its unitary irreducible representations. Since SU (2) is compact, by Theorem 5.40 all its unitary irreducible representations are finite-dimensional. Since G = SU (2) is also simply connected, by Corollary 5.43 its irreducible finite-dimensional (unitary) representations u bijectively correspond to the irreducible finite-dimensional skew-adjoint representations ρ = u of its Lie algebra g. Hence our job is to find the latter. We already encountered the basis (3.66) of the Lie algebra so(3) =∼ R3 of SO(3); the corresponding basis of the Lie algebra su(2) of SU (2) is (S1,S2,S3), where

= − 1 σ , Sk 2 i k (5.174) and the σk are the Pauli matrices given in (5.42); linear extension of the map Jk → Sk deﬁnes an isomorphism between so(3) and su(2). These matrices satisfy

[Si,S j]=εijkSk, (5.175) where εijk is the totally anti-symmetric symbol with ε123 = 1 etc., so that (5.175) comes down to [S1,S2]=S3, [S3,S1]=S2, and [S2,S3]=S1. By linearity, ﬁnding ρ is the same as ﬁnding n × n matrices

Lk = iρ(Sk) (5.176) that satisfy [Li,L j]=iεijkLk, (5.177) i.e., [L1,L2]=iL3, etc., and ∗ = . Lk Lk (5.178) It turns out to be convenient to introduce the ladder operators

L± = L1 ± iL2, (5.179) with ensuing commutation relations

[L3,L±]=±L±; (5.180)

[L+,L−]=2L3. (5.181)

Furthermore, we deﬁne the Casimir operator = 2 + 2 + 2, C L1 L2 L3 (5.182) 5.8 Irreducible representations of SU (2) 159 which, crucially, commutes with each Lk, i.e.,

[C,Lk]=0 (k = 1,2,3). (5.183)

By Schur’s lemma, in any irreducible representation we therefore must have

C = c · 1H , (5.184) where c ∈ R (in fact, c ≥ 0). We will also use the additional algebraic relations

L+L− = C − L3(L3 − 1H ); (5.185)

L−L+ = C − L3(L3 + 1H ). (5.186) ∗ = The simple idea is now to diagonalize L3, which is possible as L3 L3. Hence H = Hλ , (5.187)

λ∈σ(L3) where σ(L3) is the spectrum of L3 (which in this ﬁnite-dimensional case consists of its eigenvalues), and Hλ is the eigenspace of L3 for eigenvalue λ (i.e., if υ ∈ Hλ , then L3υ = λυ). The structure of (5.187) in irreducible representations is as follows. Lemma 5.44. Let ρ : su(2) → B(H) be a ﬁnite-dimensional skew-adjoint irreducible representation, so that (5.177) holds. Then the spectrum σ(L3) of the self- adjoint operator L3 = iρ(S3) is given by

σ(L3)={− j,− j + 1,···, j − 1, j}. (5.188)

If (5.187) is the spectral decomposition of H relative to L3, then:

1. The subspace Hλ is one-dimensional for each λ ∈ σ(L3); 2. For λ < j the operator L+ maps Hλ to Hλ+1, whereas L+ = 0 on Hj; 3. For λ > − j the operator L− maps Hλ to Hλ−1, whereas L− = 0 on H− j.

Proof. For any λ ∈ σ(L3) and nonzero υλ ∈ Hλ ,wehave:

• either λ + 1 ∈ σ(L3) and L+υλ ∈ Hλ+1 (as a nonzero vector); • or L+υλ = 0.

Indeed, (5.180) gives L3(L+υλ )=(λ + 1)L+υλ , which immediately yields the claim. Similarly, either λ − 1 ∈ σ(L3) and L−υλ ∈ Hλ−1,orL−υλ = 0. Now let λ = σ( ) = υ ∈ 0 min L3 be the smallest eigenvalue of L3, and pick some 0 λ0 Hλ0 . Since H is ﬁnite-dimensional by assumption, there must be some k ∈ N0 = N ∪{0} k+1υ = l υ = ,..., such that L+ λ0 0, whereas all vectors L+ λ0 for l 0 k are nonzero (and lie in Hλ0+l). With c deﬁned as in (5.184), it then follows from (5.185) - (5.186) that

c − λ0(λ0 − 1)=0; (5.189)

c − (λ0 + k)(λ0 + k + 1)=0. (5.190) 160 5 Symmetry in quantum mechanics

These relations imply λ0 = −k/2, so that by the above bullet points we also have

{−k/2,−k/2 + 1,...,k/2 − 1,k/2}⊆σ(L3). (5.191)

To prove equality, as in (5.188), consider the vector space

k−1 k = C · υ ⊕ C · +υ ⊕···⊕ υ ⊕ υ ⊆ H λ0 L λ0 L+ λ0 L+ λ0 H; (5.192)

k−1 (υ , +υ ,..., υ , k υ ) this is just the subspace of H with basis λ0 L λ0 L+ λ0 L+ λ0 . By the previous arguments following from (5.180), we see that the operators L+ and L− never leave H , and the same is trivially true for L3. Therefore, if ρ is irreducible, then we must have H = H (and conversely). All claims of the lemma are now trivially veriﬁed on H .

It should be clear from this proof that the actions of L+, L−, and L3 (and hence of all elements of su(2))onH = H) are ﬁxed, so that ρ is determined by its dimension

dim(H)=2 j + 1, (5.193) from which it follows that j can only take the values 0,1/2,1,3/2,.... It remains to ﬁx an inner product on H in which ρ is skew-adjoint, i.e., in which ∗ = ∗ = ∗ = ∗ = L3 L3 and L+ L− (which implies that L1 L1 and L2 L2, which jointly imply ρ(X∗)=−ρ(X) for any X ∈ g). This may be done in principle by starting with any inner product, integrating ρ to a unitary representation of SU (2), and using the construction explained at the beginning of the proof of Theorem 5.40. In practice, it is easier to just calculate: take H = Cn with n = 2 j + 1, standard inner product, and standard orthonormal basis (ul), labeled as l = 0,1,...,2 j). Then put

L u =(l − j)u ; (5.194) 3 l l L+u = (l + 1)(n − l − 1)u + ; (5.195) l l 1 L−ul = l(n − l)ul−1. (5.196)

Note that (5.195) is even formally correct for l = 2 j, since in that case n−2 j−1 = 0, and similarly, (5.196) formally holds even for l = 0. The commutation relations (5.180) - (5.181) as well as the above conditions for skew-adjointness may be explicitly veriﬁed, from which it follows that for any prescribed dimension (5.193) we have found a skew-adjoint realization of ρ. Clearly, ul = υl− j. In view of Theorem 5.40 and Corollary 5.43 we have therefore proved: Theorem 5.45. Up to unitary equivalence, any (unitary) irreducible representation of SU(2) is completely determined by its dimension n = dim(H), and any dimension n ∈ N0 = N ∪{0} occurs. Furthermore, if j is the number in (5.188), we have

n = 2 j + 1. (5.197)

Physicists typically label these irreducible representations by j (called the spin of the given representation) rather than by n,orevenbyc = j( j + 1), cf. (5.184). 5.8 Irreducible representations of SU (2) 161

Corollary 5.43 shows that one may pass from ρ(su(2)) to a unitary representation u(SU (2)), of which one may give a direct realization. For j ∈ N0/2, deﬁne Hj as the complex vector space of all homogeneous polynomials p in two variables z =(z1,z2) ( 2 j, 2 j−1 ,..., 2 j−1, 2 j) of degree 2 j. A basis of Hj is given by z1 z1 z2 z1z2 z2 , which has 2 j + 1 elements. So dim(Hj)=2 j + 1. Then consider the map

D j : SU (2) → B(Hj); (5.198)

D j(u) f (z)= f (zu). (5.199)

Clearly, D j(e) f (z)= f (z · 12)= f (z)), (5.200) so D j(e)=1, and

D j(u)D j(v) f (z)=D j(v) f (zu)= f (zuv)=D j(uv) f (z), so D j(u)D j(v)=D j(uv). Hence D j is a representation of SU (2). = − 1 We now compute L3 2 iS3 on this space. From (5.156) with u D j,wehave d it = − 1 i 0 = − 1 e 0 , L3 2 iD j − 2 i D j −it (5.201) 0 i dt 0 e t=0 so that d ∂ f (z) ∂ f (z) ( )=− 1 ( it , −it ) = 1 − . L3 f z 2 i f e z1 e z2 t=0 2 z1 z2 (5.202) dt ∂z1 ∂z2 Similarly, we obtain

∂ f (z) L+ f (z)=z1 ; (5.203) ∂z2 ∂ f (z) L− f (z)=z2 . (5.204) ∂z1 ( )= 2 j = ( )= 2 j = − Hence f2 j z z1 gives L3 f2 j jf2 j, and f0 z z2 gives L3 f0 jf0.In ( )= l 2 j−l λ = − + general, fl z z1z2 spans the eigenspace Hλ of L3 with eigenvalue j l. Since l = 0,1,...,2 j, this conﬁrms (5.188), as well as the fact that the corresponding eigenspaces are all one-dimensional. The rest is easily checked, too, except for the unitarity of the representation, for which we refer to the proof of Theorem 5.40. Finally, we return to SO(3). Either explicit exponentiation (5.165), as done for j = 1/2 in (5.168), or the above construction of D j, allows one to verify the crucial (δ)= δ ∈ = Z condition stated in Corollary 5.43, namely that D j 1Hj for D 2, which (− )= ∈ N comes down to D j 12 1Hj . This is easily seen to be the case iff j 0. Corollary 5.46. Up to unitary equivalence, each unitary irreducible representation of SO(3) is completely ﬁxed by its dimension n = 2 j +1, where j ∈ N0 (so that n = 1 for spin-0, n = 3 for spin-1, n = 5 for spin-2, ...),andeachsuchdimension occurs. 162 5 Symmetry in quantum mechanics 5.9 Irreducible representations of compact Lie groups

Because of its importance for the classical-quantum correspondence (cf. §7.1) we first reformulate the main result of the previous section (i.e. the classification the irreducible representations of SU (2)) and on that basis generalize this result to arbitrary compact Lie groups. This gives a classification of great simplicity and beauty. We already encountered the coadjoint representation (3.100) of a Lie group G on g∗, given by (x · θ)(A)=θ(x−1Ax), where x ∈ G, θ ∈ g∗, A ∈ g. The orbits under this action are called coadjoint orbits.IfG = SO(3),wehaveg =∼ R3 under the map

3 x · J ≡ ∑ xxJi → (x1,x2,x3) ≡ x, (5.205) k=1

∗ ∼ 3 where the matrices Jk are given in (3.66). Hence also g = R under the map 3 θ → (θ1,θ2,θ3) : x → ∑ θkxk . (5.206) k=1

Writing R ∈ SO(3) for a generic element x ∈ G, analogously to (5.44), we can compute the adoint action R : A → RAR−1, seen as an action on R3, through

R(x · J)R−1 =(Rx) · J. (5.207)

Using the fact that the angular momentum matrices transform as vectors, i.e.,

−1 RJiR = ∑R jiJj, (5.208) j we find that the adjoint action of SO(3) on g, seen as R3, is its defining action. In general, if g =∼ Rn and also g∗ =∼ Rn under the usual pairing of Rn and Rn through the Euclidean inner product, the coadjoint action of G on g∗, seen as an action on Rn, is given by the inverse transpose of the adjoint action on g =∼ Rn.ForSO(3) we have (R−1)T = R, so the coadjoint action of SO(3) on R3 is just its defining action, too, and hence the coadjoint orbits are the 2-spheres Sr with radius r ≥ 0. Turning to SU (2), we now make the identification of g∗ with R3 slightly differ- ently, namely by replacing the 3×3 real matrices Ji in (5.205) by the 2×2 matrices Si in (5.174), but the computation is similar: using (5.44) - (5.45), we find that the coadjoint action of u ∈ SU (2) on R3 is given by the defining action of π˜(u) ∈ SO(3), cf. (5.46). It follows that the coadjoint orbits for SU (2) are the same as for SO(3). Returning to general Lie groups G for the moment, assumed connected for simplicity, we take some coadjoint orbit O ⊂ g∗, fix a point θ ∈ O (so that O = G·θ ≡

Gθ ), and look at the stabilizer Gθ and its Lie algebra gθ . Since the derivative Ad of the adjoint action Ad of G on g—deﬁned as in (5.156)—is given by

Ad (A) : B → [A,B], (5.209) 5.9 Irreducible representations of compact Lie groups 163 it follows that the “inﬁnitesimal stabilizer” gθ is given by

gθ = {A ∈ g | θ([A,B]) = 0∀B ∈ g}. (5.210)

Consequently, the restriction of θ : g → R to gθ ⊂ g is a Lie algebra homomorphism (where R is obviously endowed with the zero Lie bracket). Consider a character χ : Gθ → T, which is the same thing as a one-dimensional unitary representation of Gθ .IfweregardT as a closed subgroup of GL1(C), its Lie algebra t is given by iR ⊂ M1(C)=C. It is conventional (at least among physicists) to take −i as the basis element of t, so that t =∼ R under −it ↔ t, so that the exponential map exp : t → T (which is the usual one), seen as a map from R to T,isgivenbyt → exp(−it). Deﬁning the derivative χ : gθ → C as in (5.156), it follows that actually

χ : gθ → iR, so that iχ maps gθ to R and is a Lie algebra homomorphism. Deﬁnition 5.47. Let G be a connected Lie group. A coadjoint orbit O ⊂ g∗ is called θ ∈ O θ = χ integral if for some (and hence all) one has |gθ i for some character χ : Gθ → T, i.e., if there is a character χ such that for each A ∈ gθ one has

d θ(A)=i χ etA . (5.211) dt |t=0 In the simplest case where G = T, the coadjoint action on t∗ is evidently trivial, so ∗ ∼ that Gθ = G = T for any θ ∈ t = R. Furthermore, any character on T takes the n ∼ form χn(z)=z , where n ∈ Z, cf. (C.351). As explained above, if t = R and hence also t∗ =∼ R, the identiﬁcation of λ ∈ t∗ with λ ∈ R is made by λ(−i) ↔ λ, where −i ∈ t.Ifχ = χn, the right-hand side of (5.211) evaluated at A = −i equals n, so that (5.211) holds iff θ = n for some n ∈ Z. Thus the integral coadjoint orbits in t∗ are the integers Z ⊂ R. Similarly, if G = Td, the characters are elements of Zd,asin

χ ( ,..., )= n1 ··· nd , (n1,...,nd ) z1 zd z1 zd (5.212) and the integral coadjoint orbits in g∗ =∼ Rd are the points of the lattice Zd ⊂ Rd. = ( ) 2 ⊂ R3 θ =( , , ) = For G SU 2 we take a coadjoint orbit Sr and ﬁx r 0 0 r .Ifr 0, then Gθ = G and (5.211) holds for the trivial character χ ≡ 1, so the orbit {(0,0,0)} > ≡ ( ) ( ) is integral. Let r 0. Then Gθr Gr consist of the pre-image of SO 2 in SU 2 under the projection π˜ in (5.46), where SO(2) ⊂ SO(3) is the group of rotations around the z-axis. This is the abelian group

T = {diag(z,z) | z ∈ T}. (5.213)

This group is isomorphic to T under diag(z,z) → z and hence its characters are n ∗ ∼ 3 given by χn(diag(z,z)) = z , where n ∈ Z. The identiﬁcation g = R is made by ∗ identifying θ ∈ g with (θ1,θ2,θ3), where θ1 = θ(Si). Putting A = S3 in (5.211), see (5.174), therefore gives r = n/2 for some n ∈ N. We conclude that the coadjoint ( ) 2 ⊂ R3 ∈ N / orbits for SU 2 are given by the two-spheres Sr with r 0 2. Similarly, for G = SO(3) the stabilizer of (0,0,r) is SO(2) =∼ T itself, and putting = 2 ∈ N A J3 in (5.211) one ﬁnds that the coadjoint orbits are the spheres Sr with r 0. 164 5 Symmetry in quantum mechanics

For any (Lie) group G, let the unitary dual Gˆ be the set whose elements are equivalence classes of unitary irreducible representations of G, where we say:

Deﬁnition 5.48. Two unitary representations ui : G → U(Hi),i= 1,2,areequiva- ∗ lent if there is unitary v : H1 → H2 such that u2(x)=vu1(x)v for each x ∈ G. The examples G = Td as well as for G = SU (2) now suggest the following theorem: Theorem 5.49. If G is a compact connected Lie group, then the unitary dual Gisˆ parametrized by the set of integral coadjoint orbits in g∗. Furthermore, there is an explicit (geometric) procedure to a construct an irreducible representation uO corresponding to such an orbit, namely by the method of geometric quantization. We will not explain this method, which would require some reasonably advanced differential geometry, but instead we outline the connection between coadjoint orbits and the well-known method of the highest weight. Let G be a compact connected Lie group and pick a maximal torus T ⊂ G. Let

WT = N(T)/T (5.214) be the corresponding Weyl group, where N(T) is the normalizer of T in G (i.e., x ∈ N(T) iff xzx−1 ∈ T for each z ∈ T). Note that all maximal tori in compact connected Lie groups are conjugate, so that the speciﬁc choice of T is irrelevant. For example, for SU (2) we take (5.213), in which case N(T) is generated by T ∼ and σ1 ∈ SU (2), so that W = S2, i.e., the permutation group on two variables. In general the Weyl group inherits the adjoint action of N(T) on T, so that WT acts on T and hence also acts on t and t∗; for SU (2) the action of the nontrivial element of WT , i.e., image [σ1] of σ1 ∈ N(T) in N(T)/T),onT is given by

[σ1](diag(z,z)) = diag(z,z), (5.215) ∼ so that its action on T = T is z → z, which gives rise to actions A →−A of WT on t ∗ and hence λ →−λ of WT on t . This is a special case of the following bijection: ∗ ∼ ∗ g /G = t /WT , (5.216) ∗ ∼ where the G-action on g is the coadjoint one; globally, one has G/Ad(G) = T/WT . ( ) 2 R3 Indeed, for SU 2 the left-hand side of (5.216) is the set of spheres Sr in , r ≥ 0, whereas the right-hand side is R/S2 (where S2 acts on R by θ →−θ). ∗ ∗ In general, a given coadjoint orbit O ⊂ g deﬁnes a Weyl group orbit OW in t as follows: O contains a point θ for which T ⊆ Gθ , and we take OW to be the orbit through θ|t. Conversely, any G-invariant inner product on g induces a decomposition

g = t ⊕ t⊥, (5.217)

∗ ∗ ⊥ ∗ which yields an extension of λ ∈ t to θλ ∈ g that vanishes on t . Let Λ ⊂ t be the set of integral elements in t∗ (as explained after Deﬁnition 5.47). Elements of Λ are called weights. Theorem 5.51 below gives a parametrization 5.9 Irreducible representations of compact Lie groups 165 ∼ Gˆ = Λ/WT , (5.218) which, restricting (5.216) to the integral part Λ ⊂ t∗, implies Theorem 5.49. Instead of with the quotient Λ/WT , one may prefer to work with Λ itself, as ∗ follows: we say that λ ∈ t is regular if w · λ for w ∈ WT iff w = e; this is the case iff λ = θ|t with Gθ = T.ForSU (2) all weights λ ∈ Z are regular except λ = 0. ∗ ∗ The set tr of regular elements of t falls apart into connected components C, called ∗ Weyl chambers, which are mapped into each other by WT .ForSU (2) one has t = (−∞,0) ∪ (0,∞), so that the Weyl chambers are (−∞,0) and (0,∞). One picks an arbitrary Weyl chamber Cd (for SU (2) this is (0,∞)) and forms Λ = Λ ∩ −, d Cd (5.219) − ∗ Λ where Cd is the closure of Cd in t . Elements of d are called dominant weights. For each element of Λ/WT there is a unique dominant weight representing it in Λ, so that instead of (5.218) we may also write what Theorem 5.51 actually gives, viz. ∼ Gˆ = Λd. (5.220)

To explain this in some detail, we need further preparation. Any (unitary) representation u : G → U(H) on some ﬁnite-dimensional Hilbert space H restricts to T, and since T is abelian, we may simultaneously diagonalize all operators u(z), z ∈ T. The operators iu (A), where A ∈ t, commute as well, so that we may decompose H = Hμ , (5.221) μ∈ΛH where ΛH ⊂ Λ contains the weights that occur in u|T , so that for each ψ ∈ Hμ ,

u(z)ψ = χμ (z)ψ (z ∈ T); (5.222) iu (Z)ψ = μ(Z)ψ (Z ∈ t), (5.223) where the character χμ : T → T corresponding to the weight μ ∈ Λ is defined as in (5.212) with μ =(n ,...,n ) and z =(z ,...,z ) ∈ T =∼ Td, where d = dim(T). 1 d 1 d ∼ For example, we have seen that the irreducible representations D j(SU (2)) on Hj = 2 j+1 C contains weights in Λ j = {− j,− j + 1,..., j − 1, j}, where j ∈ N0/2. In particular, take H = gC with some G-invariant inner product, cf. (5.148), and take u = Ad, given by Ad(x)B = xBx−1, so that Ad (A)(B)=[A,B], extended from g to gC: we write gC = g+ig and hence put Ad (A)(B+iC)=[A,B]+i[A,C], where A,B,C ∈ g. We assume that the inner product ·,·, on gC is obtained from a real inner product on g by complexification. This inner product on g may be restricted to t ⊂ g and hence induces an inner product on t∗, also denoted by ·,·,. For example, if G is semi-simple (like SU (2)), one may take the inner product on g and , = − 1 ( ( ) ( )) hence on gC to be the Cartan–Killing form A B 2 Tr Ad A Ad B , which is nondegenerate because G is semi-simple, and positive definite since G is compact. For SU (2) or SO(3) this gives the usual inner product on R3 and C3. 166 5 Symmetry in quantum mechanics

Deﬁnition 5.50. The roots of g are the nonzero weights of the adjoint representation u = Ad on H = gC. That is, writing Δ ⊂ Λ for the set of roots, we have α ∈ Δ iff α : t → R is not identically zero and there is some Eα ∈ gC such that for each Z ∈ t,

i[Z,Eα ]=α(Z)Eα , (5.224)

∗ cf. (5.223). Furthermore, subject to the choice of a preferred Weyl chamber Cd in tr , + we say α ∈ Δ is positive, denoted by α ∈ Δ ,ifα,λ > 0 for each λ ∈ Cd. + Since α,λ is real and nonzero for each α ∈ Δ and λ ∈Cd, one has either α ∈ Δ or −α ∈ Δ +, i.e., α ∈ Δ − = −Δ +. Since t is maximal abelian in g, it can also be shown that each root is nondegenerate. Writing gα = C · Eα , this gives a decomposition gC = tC gα gα . (5.225) α∈Δ + α∈Δ −

For G = SU (2), the single generator of t is S3, and taking E± = i(S1 ± iS2), we see from (5.180) that i[S3,E±]=±E±. Hence the roots are α±, given by α±(S3)=±1, and with (0,∞) as the Weyl chamber of choice, the root α+ is the positive one. We now deﬁne a partial ordering ≤ on Λ by putting μ ≤ λ iff λ − μ = ∑i niαi + for some ni ∈ N0 and αi ∈ Δ . This brings us to the theorem of the highest weight: Theorem 5.51. Let G be a connected compact Lie group. There is a parametrization ∼ Gˆ = Λd, such that any unitary irreducible representation uλ : G → Hλ in the class λ ∈ Gˆ deﬁned by a given dominant weight λ ∈ Λd has the following properties:

1. Hλ contains a unit vector υλ , unique up to a phase, such that

iuλ (Z)υλ = λ(Z)υλ (Z ∈ t); (5.226) + iuλ (Eα )υλ = 0 (α ∈ Δ ). (5.227)

2. Any other weight μ occurring in H, cf. (5.221), satisﬁes μ ≤ λ and μ = λ. The crucial point is that eqs. (5.226) - (5.227) imply

θλ (A)=iυλ ,uλ (A)υλ (A ∈ g), (5.228)

∗ ∗ where θλ ∈ g was deﬁned after (5.217) by λ ∈ Λd ⊂ t . Since each operator uλ (x) is unitary, each vector uλ (x)υλ is a unit vector, so we may form the G-orbit

Oλ = {|uλ (x)υλ uλ (x)υλ |,x ∈ G} (5.229) through |υλ υλ | in the space P1(Hλ ) of all one-dimensional projections on Hλ . ∗ Denoting the coadjoint orbit G · θλ ⊂ g by Oλ , where λ =(θλ )|t, the map

x · θλ →|uλ (x)υλ uλ (x)υλ |, (5.230)

is a G-equivariant diffeomorphism (in fact, a symplectomorphism) from Oλ to Oλ . This ampliﬁes Theorem 5.49 by making the the bijective correspondence between the set Λd of dominant weights and the set of integral coadjoint orbits explicit. 5.10 Symmetry groups and projective representations 167 5.10 Symmetry groups and projective representations

Despite the power and beauty of unitary group representations in mathematics,in the context of e.g. Wigner’s Theorem we have seen that in physics one should look at homomorphisms x → W(x), where W(x) is a symmetry of P1(H). In view of The- orems 5.4, this is equivalent to considering a single homomorphism h : G → G H , cf. (5.136). To simplify the discussion, we now drop Ua(H) from consideration and just G H = ( )/T deal with the connected component 0 U H of the identity. This restriction may be justiﬁed by noting that in what follows we will only deal with symmetries given by connected Lie groups, which have the property that each element is a product of squares x = y2. In that case, h(x)=h(y)2 is always a square and hence it cannot lie in the component Ua(H)/T (the anti-unitary case does play a role as soon as discrete symmetries are studied, such as time inversion, parity, or charge conjugation). Thus in what follows we will study continuous homomorphisms

h : G → U(H)/T, (5.231) where U(H)/T has the quotient topology inherited from the strong operator topology on U(H), as explained above. Since it is inconvenient to deal with such a quotient, we try to lift h to some map (5.137) where, in terms of the canonical projection

π : U(H) → U(H)/T, (5.232) which is evidently a group homomorphism, we have

π ◦ u = h. (5.233)

This can be done by choosing a cross-section s of π, that is, a measurable map

s : U(H)/T → U(H), (5.234) or (this doesn’t matter much) a map s : h(G)/T → U(H), such that

π ◦ s = id. (5.235)

Given h, such a cross-section s yields a map u : G → U(H) through

u = s ◦ h; (5.236) in particular, π(u(x)) = h(x). Such a lift often loses the homomorphism property, though in a controlled way, as follows. Since different choices of s must differ by a phase, and h is a homomorphism of groups, there must be a function

c : G × G → T (5.237) such that u(x)u(y)=c(x,y)u(xy)(x,y ∈ G). (5.238) 168 5 Symmetry in quantum mechanics

Indeed, since π and h are homomorphisms, we may compute

π(u(x)u(y)u(xy)−1)=π(s(h(x))π(s(h(y))π(s(h(xy)))−1 −1 = h(xy)h(xy) = h(eG)=eU(H)/T.

−1 −1 Hence u(x)u(y)u(xy) ∈ π (eU(H)/T)=T · 1H , which yields (5.238), or, more directly, ∗ c(x,y) · 1H = u(x)u(y)u(xy) . (5.239) Associativity of multiplication in G and the homomorphism property of h yield

c(x,y)c(xy,z)=c(x,yz)c(y,z), (5.240) and if we impose the natural requirement ue = 1H , we also have

c(e,x)=c(x,e)=1. (5.241)

Deﬁnition 5.52. A function c : G×G → T satisfying (5.240) and (5.241) is called a multiplier or C@2-cocycle on G (in the topological case one requires c to be Borel measurable, and for Lie groups it should in addition be smooth near the identity). The set of such multipliers, seen as an abelian group under (pointwise) operations in T, is denoted by Z2(G,T). If c takes the form

b(xy) c(x,y)= , (5.242) b(x)b(y) where b : G → T satisﬁes b(e)=1 (and is measurable and smooth near e as appropriate), then c is called a 2-coboundary or an exact multiplier. The set of trivial multipliers forms a (normal) subgroup B2(G,T) of Z2(G,T), and the quotient

Z2(G,T) H2(G,T)= (5.243) B2(G,T) is called the second cohomology group of G with coefﬁcients in T. The reason 2-coboundaries and the ensuing group H2(G,T) are interesting for our problem is as follows. Given a map x → u(x) from G to U(H) with (5.238), suppose we change u(x) to u(x) = b(x)u(x). The associated multiplier then changes to

b(x)b(y) c (x,y)= c(x,y), (5.244) b(xy) ( ) ( ) = ( , ) in that u x u y c x y uxy. In particular, a multiplier of the form (5.242) may be removed by such a transformation, and is accordingly called exact. Proposition 5.53. If H2(G,T) is trivial, then any multiplier can be removed by mod- ifying the lift u of h, and the ensuing map u : G → U(H) is a homomorphism and hence a unitary representation of G on H. In that case, any homomorphism G → U(H)/T comes from a unitary representation u : G → U(H) through (5.233). 5.10 Symmetry groups and projective representations 169

This is true by construction. By the same token, if H2(G,T) is non-trivial, then G will have projective representations that cannot be turned into ordinary ones by a change of phase (for it can be shown that any multiplier c ∈ Z2(G,T) is realized by some projective representation). Thus it is important to compute H2(G,T) for any given (physically relevant) group G, and see what can be done if it is non-trivial. To this end we present the main results of practical use. In order to state one of the main results (Whitehead’s Lemma), we need to set up a cohomology theory for g (which we only need with trivial coefﬁcients). Let Ck(g,R) be the abelian group of all k-linear totally antisymmetric maps ϕ : gk → R, with coboundary maps

δ (k) : Ck(g,R) → Ck+1(g,R); (5.245) k+1 i+ j (X0,X1,...,Xk) → ∑ (−1) ϕ([Xi,Xj],X0,...,Xˆi,...,Xˆ j,...,Xk),(5.246) i< j=1 where the hat means that the corresponding entry is omitted. For example, we have

(1) δ ϕ(X0,X1)=−ϕ([X0,X1]); (2) δ ϕ(X0,X1,X2)=−ϕ([X0,X1],X2)+ϕ([X0,X2],X1) − ϕ([X1,X2],X0).

These maps satisfy “δ 2 = 0”, or, more precisely,

δ (k+1) ◦ δ (k) = 0, (5.247) and hence we may deﬁne the following abelian groups:

Bk(g,R)=ran(δ (k−1)); (5.248) Zk(g,R)=ker(δ (k)); (5.249) Zk(g,R) Hk(g,R)= . (5.250) Bk(g,R)

Note that Bk(g,R) ⊆ Hk(g,R) because of (5.247). In particular, for k = 2 the group Z2(g,R) of all 2-cocycles on g consists of all bilinear maps ϕ : g×g → R that satisfy

ϕ(X,Y)=−ϕ(Y,X); (5.251) ϕ(X,[Y,Z]) + ϕ(Z,[X,Y]) + ϕ(Y,[Z,X]) = 0, (5.252) and its subgroup B2(g,R) of all 2-coboundaries comprises all ϕ taking the form

ϕ(X,Y)=θ([X,Y]), θ ∈ g∗. (5.253)

For example, for g = R any antisymmetric bilinear map ϕ : R2 → 0 is zero, so that

H2(R,R)=0. (5.254) 170 5 Symmetry in quantum mechanics

This has nothing to with the fact that the Lie bracket on g vanishes. Indeed, g = R2 does admit a unique nontrivial 2-cocycle, given by (half) the symplectic form, i.e.,

ϕ (( , ),( , )) = 1 ( − ). 0 p q p q 2 pq qp (5.255) Since B2(R2,R)=0, this cannot be removed, hence (5.255) generates H2(R2,R):

H2(R2,R) =∼ R. (5.256)

As far as cohomology is concerned, each Lie group and each Lie algebra has its own story, although in some cases a group of stories may be collected into a single narrative. As a case in point, a Lie algebra g is called simple when it has no proper ideals, and semi-simple when it has no commutative ideals. A Lie algebra is semi- simple iff it is a direct sum of simple Lie algebras. If a Lie group G is (semi-) simple, then so is its Lie algebra g. A basic result, often called Whitehead’s Lemma, is: Lemma 5.54. If g is semi-simple, then H2(g,R)=0.

Proof. The key point is that Ck(g,R) is a g-module under the action

k (X0 · ϕ)(X1,...,Xk)=− ∑ ϕ(X1,...,[X0,Xi],...,Xk). (5.257) i=1 For k = 2, a simple computation shows that

(X0 · ϕ)(X1,X2)=−ϕ([X0,X1],X2) − ϕ(X1,[X0,X2]) (2) (1) = δ ϕ(X0,X1,X2) − δ ϕ(X0,−)(X1,X2), (5.258)

1 where at fixed X0, the map ϕ(X0,−) is seen as an element of C (g,R). This show that g maps both B2(g,R) and Z2(g,R) onto itself. Indeed, if ϕ = δ (1)χ, then the first term in (5.258) vanishes because δ (2) ◦ δ (1) = 0, cf. (5.247), so that the right- hand side of (5.258) takes the form δ (1)(···) and hence lies in B2(g,R). Similarly, (2) (2) if δ ϕ = 0, then δ (X0 · ϕ)=0. We now use the fact that if g is semi-simple, then any finite-dimensional module is completely reducible. Consequently, as a g- module, Z2(g,R) must decompose as Z2(g,R)=B2(g,R) ⊕V, where V is some 2 g-module. Hence if ϕ ∈ V, then X0 · ϕ ∈ V. Since ϕ ∈ Z (g,R), the first term in (5.258) vanishes, whilst the second term lies in B2(g,R). Since V ∩B2(g,R)={0}, (1) we therefore have X0 · ϕ = 0, and hence δ ϕ(X0,−)(X1,X2)=0, which gives ϕ(X0,[X1,X2]) = 0, for all X0,X1,X2 ∈ g. At this point we use another implication of the semi-simplicity of g, namely [g,g]=g. It follows that ϕ = 0, whence V = {0}, from which Z2(g,R)=B2(g,R), or, in other words, H2(g,R)=0.

Theorem 5.55. Let G be a connected and simply connected Lie group. Then

H2(G,T) =∼ H2(g,R). (5.259)

Proof. This is really a conjunction of two isomorphisms: 5.10 Symmetry groups and projective representations 171

H2(G,T) =∼ H2(G,R); (5.260) H2(G,R) =∼ H2(g,R), (5.261) where R is the usual additive group, and Z2(G,R), B2(G,R), and hence H2(G,R) are deﬁned analogously to Z2(G,T) etc. The ﬁrst isomorphism is simply induced by

Z2(G,R) → Z2(G,T); (5.262) Γ (x,y) → eiΓ (x,y) ≡ c(x,y), (5.263) which preserves exactness and induces an isomorphism in cohomology (but note that (5.262) - (5.263) may not itself deﬁne an isomorphism). The isomorphism (5.261) is induced at the cochain level, too. Given a cocycle ϕ ∈ Z2(G,R), we construct a new Lie algebra gϕ (called a central extension of g) by taking gϕ = g ⊕ R as a vector space, equipped though with the unusual bracket

[(X,v),(Y,w)] = ([X,Y],ϕ(X,Y)); (5.264) the condition ϕ ∈ Z2(G,R) guarantees that this is a Lie bracket. Furthermore, gϕ is isomorphic (as a Lie algebra) to a direct sum iff ϕ ∈ B2(g,R); indeed, if (5.253) holds, then (X,v) → (X,v + θ(X)) yields the desired isomorphism gϕ → g ⊕ R. By Lie’s Third Theorem, there is a connected and simply connected Lie group Gϕ (again called a central extension of G), with Lie algebra gϕ , As a manifold, Gϕ = G×R, but the group laws are given, in terms of a function Γ : G×G → R,by

(x,v) · (y,w)=(xy,v + w +Γ (x,y)); (5.265) (x,v)−1 =(x−1,−v −Γ (x,x−1)). (5.266)

The group axioms then imply (indeed, they are equivalent to) the condition Γ ∈ 2 Z (G,R). Furthermore, two such extensions Gϕ and Gϕ are isomorphic iff the corresponding cocycles Γ and Γ are related by (5.244), and in particular, Γ ∈ B2(G,R) iff Gϕ is isomorphic (as a Lie group) to a direct product G × R, which in turn is the case iff ϕ ∈ B2(g,R). Conversely, given Γ ∈ Z2(G,R), we define the central extension Gϕ by (5.265) - (5.266), to find that the associated Lie algebra gϕ takes the above form, defining ϕ ∈ B2(g,R) through (5.264). Explicitly,

d d ϕ(X,Y)= Γ etX,esY − (X ↔ Y). (5.267) ds dt |s=t=0 Lie’s Third Theorem thus implies that the map ϕ ↔ Γ (which is not necessarily a bijection) descends to an isomorphism H2(g,R) → H2(G,R) in cohomology. Given (5.254), Theorem 5.55 immediately gives

H2(R,T)=0. (5.268)

In particular, if R is the relevant symmetry group, which is the case e.g. with time translation, by Proposition 5.53 we may restrict ourselves to unitary representations. 172 5 Symmetry in quantum mechanics

Once again, this has nothing to do with abelianness or topological triviality of R. Indeed, for G = g = R2, the Heisenberg cocycle (5.255) comes from the multiplier

i(pq −qp )/2 c0((p,q),(p ,q )) = e , (5.269) where R2 is seen as the group of translations in the phase space R2 of a particle moving on R. Accordingly, this multiplier is realized by the following projective representation of R2 on L2(R):

u(p,q)ψ(x)=e−ipq/2eixpψ(x − q). (5.270)

If R2 is the conﬁguration space of some particle, and the group R2 produces translations in the latter (i.e., of position), then the appropriate unitary representation would rather be on L2(R2) and would have trivial multiplier, viz.

u(q1,q2)ψ(x1,x2)=ψ(x1 − q1,x2 − q2). (5.271)

Similarly, G = R2, now seen as generating translations of momentum in the phase space R4 of the latter example would appropriately be represented on L2(R2) as

i(x1q1+x2q2) u(q1,q2)ψ(x1,x2)=e ψ(x1,x2). (5.272)

Corollary 5.56. Let G be a connected and simply connected semi-simple Lie group. Then H2(G,T) is trivial. Here we say that a Lie group is simple when it has no proper connected normal subgroups, and semi-simple if it has no proper connected normal abelian subgroups. For example, the “classical Lie groups” of Weyl are semi-simple, including SO(3) and SU (2), which are even simple (note that the latter does have a discrete nor- ∼ mal subgroup, namely its center {±12} = Z2). Also, products of simple Lie groups are semi-simple. However, Corollary 5.56 does not apply to SO(3), which is semi- simple but not simply connected. Here the relevant general result is: Theorem 5.57. Let G be a connected Lie group with H2(g,R)=0. Then

2 ∼ H (G,T) = π1(G). (5.273)

We need some background (cf. §C.15). For any abelian (topological) group A, the set Aˆ = Hom(A,T) (5.274) consists of all (continuous) homomorphisms (also called characters) χ : A → T; these are just the irreducible (and hence necessarily one-dimensional) unitary representations of A. This set is a group under the obvious pointwise operations

χ1χ2(a)=χ1(a)χ2(a); (5.275) χ−1(a)=χ(a)−1. (5.276) 5.10 Symmetry groups and projective representations 173

As such, the group Aˆ is called the (Pontryagin) dual of A; the Pontryagin Duality Theorem states that Aˆ =∼ A. Using Theorem 5.57 and Theorem 5.41, this gives

2 H (SO(3),T)=Z2. (5.277)

We now use Theorem 5.41 as a lemma to prove Theorem 5.57:

2 Proof. We ﬁrst state the map π1(G) → H (G,T) that will turn out to be an isomorphism. Assuming Theorem 5.41, pick a (Borel measurable) cross-section

s˜ : G → G˜ (5.278) of the canonical projection π˜ : G˜ → G = G˜/D. (5.279)

As always, this means that π˜ ◦ s˜ = idG, ands ˜ is supposed to be smooth near the ( )= identity, and chosen such thats ˜ eG eG˜ , where eG and eG˜ are the unit elements of G and G˜, respectively. Given a character χ ∈ π1(G), deﬁne cχ : G × G → T by

−1 cχ (x,y)=χ(s˜(x)s˜(y)s˜(xy) ). (5.280)

This makes sense: π˜ is a homomorphism, so that (cf. the computation below (5.238))

−1 −1 −1 π˜(s˜(x)s˜(y)s˜(xy) )=π˜(s˜(x))π˜(s˜(y))π˜(s˜(xy)) = xy(xy) = eG,

−1 and hences ˜(x)s˜(y)s˜(xy) ) ∈ ker(π˜)=D (where we identify D with π1(G), cf. Theorem 5.41). Furthermore, tedious computations show that (5.240) and (5.241) hold, so that cχ ∈ Z2(G,T). Different choices ofs ˜ lead to equivalent 2-cocycles c, and hence by taking the cohomology class [cχ ] of cχ we obtain an injective map

2 π1(G) → H (G,T); (5.281) χ → [cχ ]. (5.282)

To prove surjectivity of this map, let c ∈ Z2(G,T) and deﬁnec ˜ : G˜ × G˜ → T by

c˜(x˜,y˜)=c(π˜(x),π˜(y)). (5.283)

Conversely, we may recover c fromc ˜ and some cross-sections ˜ : G → G˜ of π˜ by

c(x,y)=c˜(s˜(x),s˜(y)). (5.284)

It follows thatc ˜ ∈ Z2(G˜,T). Theorem 5.55 implies that H2(G˜,T) is trivial, so that

c˜(x˜,y˜)=b˜(x˜y˜)/b˜(x˜)b˜(y˜), (5.285) for some function b˜ : G˜ → T satisfying b˜(e˜)=1. From (5.241), i.e., c(e,x)=1, we infer that ifx ˜ = δ ∈ D, so that π˜(δ)=e, thenc ˜(δ,y˜)=1, and hence 174 5 Symmetry in quantum mechanics

b˜(δy˜)=b˜(δ)b˜(y˜). (5.286) ˜ Takingx ˜ andy ˜ both in D, we see that b|D is a character, which we call χ. Hence

b˜(s˜(x)s˜(y)) b˜(s˜(xy)) b˜(s˜(x)s˜(y)) c(x,y)= = · b˜(s˜(x))b˜(s˜(y)) b˜(s˜(x))b˜(s˜(y)) b˜(s˜(xy)) b˜(s˜(xy)) = · cχ (x,y), (5.287) b˜(s˜(x))b˜(s˜(y)) since, using (5.286) with δ s˜(x)s˜(y)s˜(xy)−1 andy ˜ s˜(xy),wehave

b˜(s˜(x)s˜(y)) b˜(s˜(x)s˜(y)s˜(xy)−1s˜(xy)) = = b˜(s˜(x)s˜(y)s˜(xy)−1) b˜(s˜(xy)) b˜(s˜(xy)) −1 = χ(s˜(x)s˜(y)s˜(xy) )=cχ (x,y).

Thus [c]=[cχ ], and hence the map (5.281) - (5.282) is surjective.

Deﬁnition 5.58. In the situation and notation of Theorem 5.41, a unitary representation u˜ : G˜ → U(H) is called admissible if u˜(D) ⊂ T · 1H .

In that case, there is obviously a character χ ∈ Dˆ such that for each δ ∈ D we have

u˜(δ)=χ(δ) · 1H . (5.288)

Unitary irreducible representations are admissible, since Schur’s Lemma implies that, since D lies in the center of G˜, its imageu ˜(D) consists of multiples of the unit. Ifu ˜ is admissible, we obtain a homomorphism (5.231) by means of

h = π ◦ u˜ ◦ s˜, (5.289) wheres ˜ is any cross-section of π˜, cf. (5.278) - (5.279). Note that different choices s˜,s˜ are related by s˜ (x)=s˜(x)δ(x), where δ : G → D is some function, so that

h (x)=π(u˜(s˜ (x))) = π(u˜(s˜(x))u˜(δ(x))) = π(u˜(s˜(x)))π(δ(x) · 1H )=h(x).

Theorem 5.59. 1. If G is a connected Lie group with H2(g,R)=0, any homomorphism h : G → U(H)/T as in (5.231) comes from some admissible unitary representation uof˜ Gby˜ (5.289). If H is separable, then h is continuous iff u˜ is. 2. Moreover, if u˜(G˜) is super-admissible in that u˜(δ)=1H for each δ ∈ D, then u = u˜◦s˜ is a unitary representation of G, in which case h = π ◦u therefore comes from a unitary representation of G itself.

Proof. Given such a homomorphism h, pick a cross-section s : U(H)/T → U(H),as in (5.234), with associated 2-cocycle c on G given by (5.239). By Theorem 5.57 and its proof, we may assume (possibly after redeﬁning s) that there exists a character χ ∈ Dˆ and a cross-section (5.278) such that c = cχ , cf. (5.280). We then deﬁne 5.10 Symmetry groups and projective representations 175

u˜ : G˜ → B(H); (5.290) x˜ → χ(x˜· (s˜◦ π˜(x˜))−1)u(π˜(x˜)). (5.291)

Simple computations then show thatx ˜· (s˜◦ π˜(x˜))−1 ∈ D (i.e., the center of G˜), that (5.288) holds, that each operatoru ˜(x˜) is unitary, that the group homomorphism prop- ertiesu ˜(x˜)u˜(y˜)=u˜(x˜y˜) andu ˜(e˜)=1H hold, and that (5.289) is valid. As to the last equation, since π removes the term with χ in (5.291), and u = s ◦ h,wehave

π ◦ u˜ ◦ s˜(x)=π ◦ s ◦ h ◦ π˜ ◦ s˜(x)=h(x), since π ◦ s = id (on U(H)/T) and π˜ ◦ s˜ = id (on G). Ifu ˜(δ)=1H for each δ ∈ D, then cχ = 1 from (5.280), so that u(x)u(y)=uxy by (5.238). If s preserves units, or, equivalently, if he = 1H , as we always assume, we see that u is a unitary representation of G. In this case, (5.291) simply reads u˜ = s ◦ h ◦ π˜. This immediately yieldsu ˜ = u ◦ π˜, which in turn gives u = u˜ ◦ s˜. Finally, even if h is continuous, it is a priori unclear ifu ˜ is, since the cross- sections s ands ˜ appearing in the above construction typically fail to be continuous. Fortunately, since they are assumed measurable, there is no question about measurability ofu ˜, and if H is separable, continuity follows from Proposition 5.36. Corollary 5.60. If G is a connected Lie group with covering group G,˜ the formulae

u˜ = u ◦ π˜; (5.292) u = u˜ ◦ s˜, (5.293) where s˜: G → G˜ is any cross-section of the covering map π˜ : G˜ → G, give a bijective correspondence between (continuous) super-admissible unitary representations uof˜ G˜ and (continuous) unitary representations u of G, preserving irreducibility. Corollary 5.61. Any homomorphism h : SO(3) →U(H)/T as in (5.231) comes from an admissible unitary representation uofSU˜ (2) by (5.289). Moreover, h comes from a unitary representation u = u˜ ◦ sofSO˜ (3) itself iff u˜ is trivial on the center Z2. In particular, if h is irreducible, it must come from the unitary irreducible rep- = = , 1 , ,... resentations u˜ D j, where j 0 2 1 is the (half-) integer spin label. Then D j(SU (2)) is super-admissible iff j is integral, in which case it defines a unitary irreducible representation of SO(3). Indeed, the assumption H2(g,R)=0 in Theorem 5.59 is satisfied for SO(3) because of Whitehead’s Lemma 5.54. The case where H2(g,R) = 0 occurs e.g. for the Galilei group (cf. §7.6). It can be shown that H2(g,R) has finitely many gen- 2 erators, for which one finds pre-images (ϕ1,...,ϕM) in Z (g,R), with correspond- 2 ing elements (Γ1,...,ΓM) of Z (G˜,R), cf. the proof of Theorem 5.55. Of these, a subset (Γ1,...,ΓN), N ≤ M, satisfies the relation Γi(δ,x˜)=Γi(x˜,δ) for any δ ∈ D (cf. Theorem 5.41) andx ˜ ∈ G˜. This yields a map Γ : G˜ × G˜ → RN given by Γ (x˜,y˜)=(Γ1(x˜,y˜),...,ΓN(x˜,y˜)), which in turn equips the set

Gˇ = G˜ × RN, (5.294) 176 5 Symmetry in quantum mechanics with a group multiplication (x˜,v) · (y˜,w)=(x˜y˜,v + w + Γ (x˜,y˜)). We then have the following generalization of Theorem 5.59, in which a unitary representation u of Gˇ N is called admissible if u(δ,v) ∈ T · 1H for any δ ∈ D and v ∈ R . Theorem 5.62. Let G be a connected Lie group, and H a separable Hilbert space. Then any continuous homomorphism h : G → U(H)/T comes from some admissible continuous unitary representation uof˜ G.ˇ As we only apply this to the Galilei group (where N = 1), basically only for illus- trative purposes, we omit the proof. The correct (and natural) notion of equivalence of projective representations is as follows: we say that two such homomorphisms hi : G → U(Hi)/T, i = 1,2 are equivalent if there is a unitary w : H1 → H2 such that

Adw(h1(x)) = h2(x), x ∈ G, (5.295)

∗ where Adw : U(H1)/T → U(H2)/T is the map [u] → [vuv ], which is well deﬁned (here [u] is the equivalence class of u ∈ U(H) in U(H)/T under u ∼ zu, z ∈ T). This induces the following notion for Gˇ: two admissible unitary representations u˜1,u˜2 of G˜ on Hilbert spaces H1,H2 are equivalent if there is a unitary w : H1 → H2 ∗ and a map b : Gˇ → T such that wu1(xˇ)w = b(xˇ)u2(xˇ), for anyx ˇ ∈ Gˇ. It can be shown that such a map b always comes from a character χ : G˜ → T through b(x˜,v)=χ(x˜).

To close this long and difficult section, in relief it should be mentioned that the above theory vastly simplifies if H is finite-dimensional. By Theorem 5.40, this is true, for example, if G is compact and u is irreducible. Suppose u : G → U(H) is merely a projective unitary representation of G, so that instead of (5.157) one has

[u (X),u (Y)] = u ([X,Y]) + iϕ(X,Y) · 1H , (5.296) where ϕ is given by (5.267). Taking the trace yields i ϕ(X,Y)= Tr (u ([X,Y])), (5.297) n where n = dim(H) < ∞. We may deﬁne a linear function θ : g → R by i θ(X)= Tr (u (X)), (5.298) n so that ϕ(X,Y)=θ([X,Y]), cf. (5.253), and hence we may remove ϕ by redeﬁning

u˜ (X)=u (X)+iθ(X) · 1H , (5.299) which satisfies (5.157) - (5.158). Hence by Corollary 5.43 the mapu ˜ exponentiates to a unitary representationu ˜ of the universal covering group G˜ of G; it should be checked from the values ofu ˜ on D ifu ˜ also defines a unitary representation of G. This argument shows that finite-dimensional projective unitary representations of Lie groups always come from unitary representations of the covering group. 5.11 Position, momentum, and free Hamiltonian 177 5.11 Position, momentum, and free Hamiltonian

The three basic operators of non-relativistic quantum mechanics are position, denoted q, momentum, p, and the free Hamiltonian h0. Assuming for simplicity that the particle moves in one dimension, these are informally given on H = L2(R) by

qψ(x)=xψ(x); (5.300) d pψ(x)=−ih¯ ψ(x); (5.301) dx h¯ 2 d2 h ψ(x)=− ψ(x), (5.302) 0 2m dx2 where m is the mass of the particle under consideration. We put h¯ = 1 and m = 1/2. The issue is that these operators are unbounded; see §B.13. In general, quantum- mechanical observables are supposed to be represented by self-adjoint operators, and examples like (5.300) - (5.302) show that these may not be bounded. The Hellinger–Toeplitz Theorem B.68 then shows that it makes no sense to try and extend the above expressions to all of L2(R), so we have to live with the fact that some crucial operators a : D(a) → H are merely deﬁned on a dense subspace D(a) ⊂ H. Each such operator has an adjoint a∗ : D(a∗) → H, whose domain D(a∗) ⊂ H consists of all ψ ∈ H for which the functional ϕ →ψ,aϕ is bounded on D(a), and hence (since D(a) is dense in H) can be extended to all of H by continuity through the unique “Riesz–Frechet´ vector” χ for which ψ,aϕ = χ,ϕ. Writing χ = a∗ψ, for each ψ ∈ D(a∗) and ϕ ∈ D(a) we therefore have

a∗ψ,ϕ = ψ,aϕ. (5.303)

Assuming that D(a) is dense in H, we say that a is self-adjoint, written a∗ = a,if

aϕ,ψ = ϕ,aψ, (5.304) for each ψ,ϕ ∈ D(a) and D(a∗)=D(a). A self-adjoint operator a is automatically closed, in that its graph G(a)={(ψ,aψ) | ψ ∈ D(a)} is a closed subspace of the Hilbert space H ⊕ H (indeed, the adjoint of any densely defined operator is closed, see Proposition B.72). In practice, self-adjoint operators often arise as closures of essentially self-adjoint operators a, which by definition satisfy a∗∗ = a∗. Equiva- lently, such an operator is closable, in that the closure of its graph is the graph of some (uniquely defined) operator, called the closure a− of a, and furthermore this closure is self-adjoint, so that a− = a∗.Ifa is closable, the domain D(a−) of its closure consists of all ψ ∈ H for which there exists a sequence (ψn) in D(a) such − − that ψn → ψ and aψn converges, on which we define a by a ψ = limn aψn. The simplest case is the position operator. Theorem 5.63. The operator q is self-adjoint on the domain D(q)={ψ ∈ L2(R) | dxx2|ψ(x)|2 < ∞}. (5.305) R 178 5 Symmetry in quantum mechanics

See Proposition B.73 for the proof. To give a convenient domain of essential self- adjointness (also for the other two operators), we need a little distribution theory. Deﬁnition 5.64. The Schwartz space S (R) (whose elements are functions of rapid decrease) consist of all smooth function f : R → C for which each expression

n (m) f n,m = sup{|x f (x)|,x ∈ R}, (5.306) where f (m) is the m’th derivative of f , is ﬁnite. The topology of S (R) is given by saying that a sequence (or net) fλ converges to f iff fλ − f n,m → 0 for all n,m ∈ N.

Each ·n,m happens to be a norm, but positive definiteness is nowhere used in the theory below (which therefore works for families of seminorms, which satisfy the axioms of a norm expect perhaps for positive definiteness). Since there are countably many such (semi)norms defining the topology, we may equivalently say that S (R) is a metric space defined by ∞ − ( , )= −n f g n,m . d f g ∑ 2 + − (5.307) n,m=0 1 f g n,m

Indeed, S (R) is complete in this metric. A typical element is f (x)=exp(−x2). Deﬁnition 5.65. A tempered distribution is a continuous linear map ϕ : S (R) → C. The space of all such maps, equipped with the topology of pointwise convergence

(i.e., ϕλ → ϕ iff ϕλ ( f ) → ϕ( f ) for each f ∈ S (R)) is denoted by S (R). It can be shown that (because of the metrizability of S (R)) continuity is the same as sequential continuity, i.e., some linear map ϕ : S (R) → C belongs to S (R) iff limN ϕ( fN)=ϕ( f ) for each convergent sequence fN → f in S (R). Like S (R), the tempered distributions S (R) form a (locally convex) topological vector space, that is, a vector space with a topology in which addition and scalar multiplication are continuous. The topology of S (R) is given by a family of seminorms, namely

ϕ f = |ϕ( f )|, f ∈ S (R), and hence a simple way to prove that ϕ ∈ S (R) is to find some (n,m) for which |ϕ( f ))|≤C f n,m for each f ∈ S (R), since in that case fN → f , which means that fN − f n,m → 0 for all n,m ∈ N, certainly implies that ϕ( fN) → ϕ( f ), so that ϕ is continuous. For example, the evaluation maps δx defined by δx( f )= f (x) are continuous (take n = m = 0). Similarly, each finite measure on R defines a tempered distribution. Taking the (0,m) seminorm shows that the maps f → f (m)(x) for fixed m ∈ N and x ∈ R are tempered distributions. A less obvious example (defining a so-called Gelfand triple) is as follows: Proposition 5.66. We have continuous dense inclusions

S (R) ⊂ L2(R) ⊂ S (R), (5.308) where the second inclusion identiﬁes ϕ ∈ L2(R) with the map f →ϕ, f = dxϕ(x) f (x). (5.309) R 5.11 Position, momentum, and free Hamiltonian 179

Proof. As vector spaces, the ﬁrst inclusion is obvious. For f ∈ S (R) we estimate 2 = | ( )|·| ( )|≤ f 2 dx f x f x f 1 f ∞; (5.310) R (1 + x2)| f (x)| 1 f 1 = dx ≤ dy (1 + m 2 ) f ∞ R 1 + x2 R 1 + y2 x ≤ π( f 0,0 + f 2,0), (5.311) so that, noting that ·0,0 = ·∞,wehave 2 ≤ π( + ) . f 2 f ∞ f 2,0 f ∞ (5.312)

Hence fλ → f in S (R), which incorporates the conditions fλ − f 0,0 → 0 and fλ − f 2,0 → 0, implies fλ − f 2 → 0. This shows that the first inclusion in (5.308) is continuous. Density may be proved in two steps. First, take some ∈ ∞(− , ) ( )= fixed positive function h Cc 1 1 with the property dxh x 1, and define ( )= ( ) ∈ ∞(R) δ → ∞ hn x nh nx , so that informally hn Cc converges to a -function as n . 2 For each ψ ∈ L (R), we consider the convolution hn ∗ ψ, where for suitable f ,g, f ∗ g(x) ≡ dy f(x − y)g(y). (5.313) R

∞ 2 Then hn ∗ ψ ∈ C (R) ∩ L (R) and, from elementary analysis, hn ∗ ψ − ψ→0. ψ ∈ (R) ∗ ψ ∞(R) S (R) Second, for Cc , the functions hn lie in Cc and hence in . 2 2 Since Cc(R) is dense in L (R) by Theorem B.30, for ψ ∈ L (R) and ε > 0we can ﬁnd ϕ ∈ Cc(R) such that ψ − ϕ < ε/2, and (as just shown) ﬁnd n such that 2 ϕ − ϕn < ε/2, whence ψ − ϕn < ε. This proves that S (R) is dense in L (R). The second inclusion is continuous by Cauchy–Schwarz, which gives

|ϕ( f )|≤ϕ2 f 2, to be combined with (5.312). It should be noted that also the second inclusion in (5.308) is indeed an injection, i.e., that ϕ( f )=0 for each f ∈ S (R) implies ϕ = 0 in L2(R); this is true because S (R) is dense in L2(R), plus the standard fact that, in any Hilbert space H,ifϕ, f = 0 for all f in some dense subspace of H, then ϕ = 0. Finally, the fact that L2(R) is dense in the seemingly huge space S (R) follows from the even more remarkable fact that S (R) is dense in S (R). On top of the χ ∈ ∞(R) χ( )= functions hn just defined, also employ a function Cc such that x 1on (−1,1), and define χn(x)=χ(x/n), so that informally limn→∞ χ(x)=1 (as opposed to the hn, which converge to a δ-function as n → ∞). If for any g ∈ S (R) and any ϕ ∈ S (R) we define gϕ as the distribution that maps f ∈ S (R) to ϕ( fg), and similarly define g ∗ ϕ as the distribution that maps f to ϕ(g ∗ f ), we may define a ϕ = ∗ (χ ϕ) sequence of distributions n hn n . From the point of view of (5.308), these correspond to functions ϕn ∈ S (R) in the sense that ϕn( f )= dxϕn(x) f (x), where f ∈ S (R). Using similar analysis as above, it then follows that for any f ∈ S (R) we have ϕn( f ) → ϕ( f ), so that ϕn → ϕ in S (R). 180 5 Symmetry in quantum mechanics

For our purposes, the point of all this is that we can define generalized derivatives of (tempered) distributions, and hence, because of (5.308), of functions in L2(R). Definition 5.67. For ϕ ∈ S (R) and m ∈ N, the m’th generalized derivative ϕ(m) is defined by ϕ(m)( f )=(−1)mϕ( f (m)). (5.314) The idea is that under (5.308) this is an identity if ϕ ∈ S (R) (partial integration). Like the constructions at the end of the proof of Proposition 5.66, this is a special case of a more general construction: whenever we have a continuous linear map T : S (R) → S (R), we obtain a dual continuous linear map T : S (R) → S (R) defined by T ϕ = ϕ ◦ T, i.e.,

(T ϕ)( f )=ϕ(T( f )). (5.315)

Sometimes a slight change in the deﬁnition (as in (5.314), or as in the Fourier transform below) is appropriate so that the restriction of T to S (R) coincides with T. Theorem 5.68. The momentum operator p = −id/dx is self-adjoint on the domain

D(p)={ψ ∈ L2(R) | ψ ∈ L2(R)}, (5.316) where the derivative ψ is taken in the distributional sense (i.e., letting ψ ∈ S (R)). Proof. We ﬁrst show that p is symmetric, or p ⊆ p∗. This comes down to

ψ ,ϕ = −ψ,ϕ , (5.317) for each ψ,ϕ ∈ D(p), where both derivates are “generalized”. The most elegant proof (though perhaps not the shortest) uses the Sobolev space H1(R), which equals D(p) as a vector space, now equipped, however, with the new inner product

ψ,ϕ(1) = ψ,ϕ + ψ ,ϕ , (5.318) with both inner products on the right-hand side in L2(R); the associated norm is ψ2 = ψ2 + ψ 2. (1) (5.319) Similar to the Gelfand triple (5.308), we have dense continuous inclusions

S (R) ⊂ H1(R) ⊂ S (R), (5.320) with analogous proof. All we need for Theorem 5.68 is the ﬁrst inclusion of the 1 ∞ 1 triple (5.320): for ψ ∈ H (R) we now have hn ∗ ψ ∈ C (R) ∩ H (R) as well as 1 2 hn ∗ ψ → ψ in H (R), both of which follow from the L -case plus the identity

(hn ∗ ψ) = hn ∗ ψ . (5.321) χ 2 χ ψ → ψ χ ψ → Using the same cutoff function as in the L case, we have n and n 2 2 1 0inL (R), so that (χnψ) → ψ in L (R) and hence χnψ → ψ also in H (R). 5.11 Position, momentum, and free Hamiltonian 181

ψ = ∗(χ ψ) ∞(R) S (R) Furthermore, the functions n hn n lie in Cc and hence in ; using 1 1 the above facts we obtain ψn → ψ in H (R). In sum, for each ψ ∈ H (R) we can (ψ ) S (R) ψ → ψ ψ → ψ 2(R) ﬁnd a sequence n in such that n and n in L . Hence

ψ,ϕ = limψn,ϕ = −limψ ,ϕ = −ψ ,ϕ . (5.322) n n n

For the converse, let ψ ∈ D(p∗), so that by deﬁnition for each ϕ ∈ D(p) we have

p∗ψ,ϕ = ψ, pϕ = −iψ,ϕ . (5.323)

Since S (R) ⊂ D(p), this is true in particular for each ϕ ∈ S (R), in which case the right-hand side equals −iψ (ϕ), where the derivative is distributional. But this equals p∗ψ,ϕ and so the distribution −iψ is given by taking the inner product with p∗ψ ∈ L2(R). Hence −iψ = p∗ψ ∈ L2(R), and in particular ψ ∈ L2(R),so that ψ ∈ D(p). This proves that D(p∗) ⊆ D(p), and since from the ﬁrst step we have the oppositie inclusion, we ﬁnd D(p∗)=D(p) and p∗ = p.

2 2 For the free Hamiltonian h0 = −Δ with Δ = d /dx , we similarly have:

Theorem 5.69. The free Hamiltonian h0 = −Δ is self-adjoint on the domain

D(Δ)={ψ ∈ L2(R) | ψ ∈ L2(R)}, (5.324) where the double derivative ψ is taken in the distributional sense. Although this may be proved in an analogous way, such proofs are increasingly burdensome if the number of derivatives gets higher. It is easier to use the Fourier transform (which also provided an alternative way of proving Theorem 5.68). Theorem 5.70. The formulae ∞ dx fˆ(k)= √ e−ikx f (x); (5.325) −∞ π 2 ∞ dk fˇ(x)= √ eikx f (k), (5.326) −∞ 2π are rigorously deﬁned on S (R),L2(R), and S (R), and provide continuous isomorphisms of each of these spaces. Furthermore, (5.326) is inverse to (5.325), i.e.

fˇˆ = fˆˇ = f , (5.327) so that we may (and often do) write fˆ = F ( f ) and fˇ = F −1( f ),orf= F −1( fˆ). In all three cases we have the identities (in a distributional sense if appropriate)

F (xn f (m))(k)=(id/dk)n(ik)mF ( f )(k). (5.328)

Finally, as a map F : L2(R) → L2(R) the Fourier transform is unitary, so that

ψˆ ,ϕˆ = ψ,ϕ. (5.329) 182 5 Symmetry in quantum mechanics

See §C.15 for further discussion. For example, we have

D(p)={ψ ∈ L2(R) | k · ψˆ (k) ∈ L2(R)}; (5.330) D(Δ)={ψ ∈ L2(R) | k2 · ψˆ (k) ∈ L2(R)}. (5.331)

Thus we may now reformulate Theorems 5.68 and 5.69 as follows: Theorem 5.71. The momentum operator p is self-adjoint on the domain (5.330). The free Hamiltonian h0 = −Δ is self-adjoint on the domain (5.331).

Proof. Denoting multiplication by xn by the symbol kn,wehave

p = F −1kF ; (5.332) Δ = −F −1k2F . (5.333)

Hence the theorem follows from Proposition B.73 and unitarity of the Fourier transform F (plus the little observation that if a = a∗ on D(a) ⊂ H and u : H → K is unitary, then b = uau∗ is self-adjoint on D(b)=uD(a) ⊂ K).

Much is known about regularity properties of functions in such domains, e.g.,

D(p) ⊂ C0(R); (5.334) (Δ) ⊂ (1)(R). D C0 (5.335) These are the most elementary cases of the famous Sobolev Embedding Theorem. If ψ ∈ D(p), then k → (1+k2)1/2ψˆ (k) is in L2(R), so applying Holder’s¨ inequality (B.15) with p = q = 2to f (k)=(1+k2)1/2ψˆ (k) and g(k)=(1+k2)−1/2, which is in L2(R), too, gives ψˆ ∈ L1(R). The Riemann–Lebesgue Lemma (see §C.15) then 2 yields ψ ∈ C0(R). To prove (5.335), one uses (1 + k ) rather than its square root. Finally, we give a common domain of essential self-adjointness for q, p, and h0.

Proposition 5.72. The operators q, p, and h0 are essentially self-adjoint on S (R).

Proof. We see from (5.332) that the cases of p and q are similar, so we only explain the case of q. Denoting the operator of multiplication by x on the domain S (R) by ( ∗)= ( ) q0, as in the proof of Proposition B.73 it is easy to see that D q0 D q . Fourier- transforming, the fact that S (R) is dense in H1(R) (cf. the proof of Theorem 5.68) ( −)= ( ) ( ∗)= ( −) ∗ − shows that D q0 D q ,so that D q0 D q0 . The actions of q0 and q0 obvi- ∗ = − ously being given by multiplication by x in both cases, we have q0 q0 . The proof for h0 is similar; in the second step we now use the fact that S (R) is dense in H2(R), deﬁned as D(Δ), as in (5.324), but now seen as a Hilbert space in the inner product ψ,ϕ(2) = ψ,ϕ+ψ ,ϕ , with corresponding norm given by ψ2 = ψ2 + ψ 2 (2) . This is proved just as in the case of a single derivative. We also say that S (R) is a core for the operators in question. For example, the canonical commutation relations [q, p]=ih¯ · 1H rigorously hold on this domain. 5.12 Stone’s Theorem 183 5.12 Stone’s Theorem

We now come to a central result on symmetries in quantum mechanics “explaining” the Hamiltonian. Recall that a continuous unitary representation of R (as an additive group) on a Hilbert space H is a map t → ut , where t ∈ R and each ut ∈ B(H) is unitary, such that the associated map R × H → H, (t,ψ) → ut ψ, is continuous, and

usut = us+t , s,t ∈ R; (5.336)

u0 = 1H ; (5.337)

limut ψ = ψ (t ∈ R, ψ ∈ H). (5.338) t→0 These conditions imply

limut ψ = usψ (s,t ∈ R,ψ ∈ H). (5.339) t→s Note that according to Proposition 5.36 continuity may be replaced by weak measurability. Probably the simplest nontrivial example is given by H = L2(R) and

ut ψ(x)=ψ(x −t). (5.340)

To prove (5.338), we use a routine ε/3 argument. We ﬁrst prove (5.338) for ψ ∈ Cc(R), where it is elementary in the sup-norm, i.e., limt→0 ut ψ − ψ∞ = 0 by continuity and hence (given compact support) uniform continuity of ψ. But then ψ2 ≤| |ψ ⊂ R the (ugly) estimate 2 K ∞, where K is any compact set containing the support of ψ, also yields limt→0 ut ψ − ψ2 = 0. Hence for ε > 0 we may ﬁnd

δ > 0 such that ut ψ − ψ2 < ε/3 whenever |t| < δ. For general ψ ∈ H,weﬁnd

ψ ∈ Cc(R) such that ψ − ψ < ε/3, and, using unitarity of ut , estimate

ut ψ − ψ ≤ut ψ − ut ψ + ut ψ − ψ + ψ − ψ ≤ ε/3 + ε/3 + ε/3 = ε.

In the context of quantum mechanics, physicists formally write

−ita ut = e , (5.341) where a is usually thought of as the Hamiltonian of the system, although in the previous example it is rather the momentum operator. In any case, we avoid the notation h instead of a here, partly in order to rightly suggest far greater generality of the construction and partly to avoid confusion with the notation in §B.21; if his the Hamiltonian, one would have a = h/h¯ in (5.341). Mathematically speaking, if a is self-adjoint, eq. (5.341) is rigorously deﬁned by Theorem B.158, where

et (x)=exp(−itx). (5.342) 184 5 Symmetry in quantum mechanics

Conversely, given a continuous unitary representation t → ut of R on H, one may attempt to deﬁne an operator a by specifying its domain and action by u − 1 D(a)= ψ ∈ H | lim s ψ exists ; (5.343) s→0 s u − 1 aψ = i lim s ψ (ψ ∈ D(a)). (5.344) s→0 s Stone’s Theorem makes this rigorous, and even turns the passage from the generator a to the unitary group t → ut (and back) into a bijective correspondence.

Theorem 5.73. 1. If a : D(a) → H is self-adjoint, the map t → ut defined by (5.341), which is rigorously defined by Proposition B.159 with (5.342), defines a continuous unitary representation of R on H. 2. Conversely, given such a representation, the operator a defined by (5.343) - (5.344) is self-adjoint; in particular, D(a) is dense in H. 3. These constructions are mutually inverse. Proof. We use the setting of §B.21, so that b is the bounded transform of a. 1. Eqs. (5.336) - (5.337) are immediate from Theorem B.158, which also yields ϕ ∈ ∗( ) unitarity of each operator ut . To prove (5.338) we first take Cc b H, which means that ϕ is a finite linear combinations of vectors of the type ϕ = h(a)ψ, where h ∈ Cc(σ(a)) and ψ ∈ H. Using (5.342) and (B.573), we have

(K) ut ϕ − ϕ≤et h − h∞ψ≤h∞et − 1K∞ ψ, (5.345)

where K is the (compact) support of h in σ(b). Since the exponential function is uniformly convergent on any compact set, this gives limt→0 ut ϕ − ϕ = 0. Taking finite linear combinations of such vectors ϕ gives the same result for any ϕ ∈ ∗( ) ∗( ) Cc b H (with an extra step this could have been done on C0 b H, too). Thus for ε > 0 we can find δ > 0 so that ut ϕ −ϕ < ε/3 whenever |t| < δ.For ψ ∈ ϕ ∈ ∗( ) ϕ − ψ < ε/ general H,wefind C0 b H such that 3, and estimate

ut ψ − ψ ≤ut ψ − ut ϕ + ut ϕ − ϕ + ϕ − ψ ≤ ε/3 + ε/3 + ε/3 = ε,

since ut ψ − ut ϕ = ψ − ϕ by unitarity of ut . This is equivalent to (5.338). 2. For any ψ ∈ H and n ∈ N, deﬁne ψn ∈ H by ∞ −ns ψn = n dse usψ, (5.346) 0

either as a Riemann-type integral (whose approximants converge in norm) or as ϕ → ∞ −ns ψ,ϕ a functional n 0 dse us , which is obviously continuous and hence is represented by a unique vector ψn ∈ H. Then simple computations show that

us − 1 lim ψn = n(ψn − ψ), s→0 s 5.12 Stone’s Theorem 185

so that ψn ∈ D(a). The proof that ψn → ψ starts with the elementary estimate ∞ −ns ψn − ψ≤n dse usψ − ψ, 0 ∞ δ ···+ ∞ ··· δ > in which we split up the 0 as 0 δ , where 0. Using strong continuity of the map t → ut , i.e., (5.338), for any n the ﬁrst integral vanishes as δ → 0. In the second integral we estimate usψ − ψ≤2ψ and take the limit n → ∞. Thus ψn → ψ, so that D(a) is dense in H. To prove self-adjointness of a, we need a tiny variation on Theorem B.93: Lemma 5.74. Let a be symmetric. Then a is self-adjoint (i.e. a∗ = a) iff

ran(a + i)=ran(a − i)=H. (5.347)

Proof. We only need the implication from (5.347) to a∗ = a (but the converse immediately follows from Theorem B.93). So assume (5.347). For given ψ ∈ D(a∗) there must then be a ϕ ∈ H such that (a∗ − i)ψ =(a − i)ϕ. Since a is symmetric, we have D(a) ⊂ D(a∗),soψ − ϕ ∈ D(a∗), and (a∗ − i)(ψ − ϕ)=0. But ker(a∗ − i)=ran(a + i)⊥,soker(a∗ − i)=0. Hence ψ = ϕ, and in particular ψ ∈ D(a) and hence D(a∗) ⊂ D(a). Since we already know the opposite inclusion, we have D(a∗)=D(a). Given symmetry, this implies a∗ = a.

Continuing the proof of Theorem 5.73.2, symmetry of a easily follows from its ∗ = −1 = ψ,ϕ ∈ ( ) deﬁnition, combined with the property ut ut u−t . Indeed, for D a , the weak limit s → 0 below exists by deﬁnition of D(a), cf. (5.343), whence:

u − 1 u− − 1 ϕ,aψ = i limϕ, s ψ = −i lim s ϕ,ψ = aϕ,ψ. s→0 s s→0 −s

To prove that ran(a − i)=H, we compute (a − i)ψ1 = −iψ, with ψ1 deﬁned by = (− )= (5.346) withn 1. The property ran i H is proved in a similar way: now 0 s deﬁne ψ˜1 = −∞ dse usψ and obtain (a + i)ψ˜1 = iψ. Thus Lemma 5.74 applies. 3. Bijectivity has two directions: a → ut → a and ut → a → ut .

• Given a and hence (5.341) defining ut , we change notation from a to a in (5.343) - (5.344) and need to show that a = a. Denoting the restriction of ∗( ) ⊆ a to the domain Cc b by a0, we first show that a0 a . The technique to prove this is similar to the argument around (5.345). We initially assume that ϕ ∈ ( )= ∗( ) ϕ = ( )ψ ∈ (σ( )) D a0 Cc b H takes the form h a for some h Cc a and ψ ∈ H. Just a trifle more complicated than (5.345), using (5.342), (B.573), and unitarity of ut , we estimate: ! ! ! ! ! ! ! ! !ut+sϕ − ut ϕ ! !esh − h ! ! + ia0ut ϕ! ≤ ! + i · idσ(T)h! ψ s s ∞ ! ! ! !(K) !es − 1K ! ≤ ! + i · idK! h∞ψ, s ∞ 186 5 Symmetry in quantum mechanics

so that by deﬁnition of the (strong) derivative we obtain

dut ut+sϕ − ut ϕ ϕ = lim = −iaut ϕ, (5.348) dt s→0 s initially for any ϕ of the said form h(a)ψ, and hence, taking ﬁnite sums, for any ϕ ∈ D(a0). The existence of this limit shows that, on the assumption

ψ ∈ D(a0),wehaveψ ∈ D(a ), and we also see that a = a on D(a0),or,in

other words, that a0 ⊆ a . Since a is self-adjoint (by part 2 of the theorem) and − ⊆ hence closed, we have a0 a . Since a0 is essentially self-adjoint by Theorem B.159, this gives a ⊆ a . Taking adjoints reverses the inclusion, and since both operators are self-adjoint this gives a = a . • Given ut and hence (5.343) - (5.344) deﬁning a, we change notation from ut = to ut in (5.341) and need to show that ut ut . Indeed, let

ψt = ut ψ, (5.349) ψ = ψ ψ ∈ ( ) and similarly t ut .If D a , then by deﬁnition of a we have

dψt ut+s − ut us − 1H i = i lim ψ = i lim ut ψ = aψt , (5.350) dt s→0 s s→0 s ψ ∈ ( ) ψ / = ψ ψ ψ which also shows that t D a . Similarly, id t dt a t , so that t and t satisfy the same differential equation with the same initial condition ψ(0) =(ψ(0)) = ψ.

ψ = ψ −ψ Now consider ˆt t t , which once again satisﬁes the same equation (i.e., (0) (0) idψˆt /dt = aψˆt ), but this time with initial condition ψˆ0 = ψ − (ψ ) = ψ − ψ = 0. The key point is that any solution ψˆt of this equation has the property ψˆt = ψˆ0 for any t ∈ R, since by symmetry of a, d d ψˆ 2 = ψˆ ,ψˆ = −i(ψˆ ,aψˆ −aψˆ ,ψˆ )=0. dt t dt t t t t t t ψ ψ = ψ = ψ = For our speciﬁc ˆt we have ˆ0 0 and hence t t , that is, ut ut .

Corollary 5.75. With t → ut and a defined and related as in Theorem 5.73, if ψ ∈ D(a), for each t ∈ R the vector ψt defined by (5.349) lies in D(a) and satisfies dψ aψ = i t , (5.351) t dt

(0) whence t → ψt is the unique solution of (5.351) with initial value ψ = ψ. This follows from the proof of part 3 of Theorem 5.73. With a = h/h¯ (as above), this is just the famous time-dependent Schrodinger¨ equation dψ hψ = ih¯ t . (5.352) t dt Notes 187 Notes

§5.1. Six basic mathematical structures of quantum mechanics Wigner’s Theorem was first stated by von Neumann and Wigner (1928), but the first proof appeared in Wigner (1931). See Bonolis (2004) and Scholz (2006) for some history. Instead of working with P1(H) with the bilinear trace form expressing the transition probabilities, one may also formulate and prove Wigner’s Theorem in terms of the projective Hilbert space PH equipped with the Fubini–Study metric, in which case the relevant symmetries may be defined geometrically as isometries. See Freed (2012) for this proof, as well as Brody & Hughston (2001) for the underlying geometry. Kadison’s Theorem may be traced back from Kadison (1965). See also Moretti (2013). Ludwig symmetries go back to Ludwig (1983); see also Kraus (1983). Our approach to von Neumann symmetries was inspired by Hamhal- ter (2004), and has a large pedigree in quantum logic. Bohr symmetries were intro- duced in Landsman & Lindenhovius (2016), where Theorem 5.4.6 was also proved.

§5.2. The case H = C2 This material is partly based on Simon (1976). The covering map (5.46) has a nice geometric description: if Σ = C ∪{∞} is the Riemann sphere, we have the well-known stereographic projection

=∼ S2 → Σ; (5.353) x + iy (x,y,z) → . (5.354) 1 − z If u ∈ SU (2) is given by (5.43), then the associated Mobius¨ transformation

αz + β z → −βz + α is a bijection of Σ, whose associated transformation of S2 is the rotation R = π˜(u). §5.3. Equivalence between the six symmetry theorems Most proofs may be also found in Cassinelli et al (2004) or Moretti (2013). §5.4. Proof of Jordan’s Theorem Our proof of Jordan’s Theorem is taken from Bratteli & Robinson (1987); see also Thomsen (1982) for a simpliﬁcation of the purely algebraic step (which we delegated to Theorem C.175), originally proved by Jacobson & Rickart (1950). §5.5. Proof of Wigner’s Theorem There are many proofs of Wigner’s Theorem, none of them really satisfactory (in this respect the situation is similar to Gleason’s Theorem). Our proof follows Simon (1976), who in turn relies on Bargmann (1964) and Hunziker (1972). The proof in Cassinelli et al (2004) seems cleaner, but their proof of the additivity of their operator Tω is not easy to follow. For a geometric approach see Freed (2012). 188 5 Symmetry in quantum mechanics

If dim(H) ≥ 3, the conclusion of Wigner’s Theorem follows if W merely preserves orthogonality (Uhlhorn, 1963). See also Cassinelli et al (2004). This, in turn, has been generalized in various directions, e.g. to indefinite inner product spaces (Molnar,´ 2002) as well as to certain Banach spaces, where one says that x is orthogonal to y if for all λ ∈ C one has x + λy≥x (Blanco & Turnsek,ˇ 2006). §5.6. Some abstract representation theory Among numerous books on representation theory, our personal favourite is Barut & Raçka (1977), and also Gaal (1973) and Kirillov (1976) are classics at least for the abstract theory. An interesting recent paper on the unitary group on infinite- dimensional Hilbert space is Schottenloher (2013). §5.7. Representations of Lie groups and Lie algebras This section was inspired by Hall (2013) and Knapp (1988). For Lie’s Third The- orem, see, for example, Duistermaat & Kolk (2000), §1.14. To obtain Theorem 5.41, consider the canonical projection π˜ : G˜ → G and define D = π˜ −1({e}). This is a discrete normal subgroup of G˜, and it is an easy fact that a discrete normal subgroup of any connected topological group must lie in its center. Note that a discrete subgroup of the center of G˜ is automatically normal. The exponentiation problem for skew-adjoint representations of g is consider- ably more complicated than in finite dimension. Let H be an infinite-dimensional Hilbert space with dense subspace D and let ρ : g → L(D,H) be a linear map, where L(D,H) is the space of linear maps from L to H. We say that ρ is a skew-adjoint representation of g if (i): D is invariant under u (g), (ii): the commutation relations (5.157) hold on D, and (i): each iρ(A) is essentially self-adjoint on D. For example, we have seen that if u : G → U(H) is a unitary representation, then the construction

ρ(A)=u (A), deﬁned on the Garding˚ domain D = DG, ﬁts the bill. Conversely, additional conditions are needed for ρ to exponentiate to a unitary representation. The best-known of those is Nelson’s criterion: if, given a skew-adjoint representation ρ → ( , ) Δ = ∑dim(g) ρ( )2 : g L D H , the Nelson operator or Laplacian k=1 Tk is essentially self-adjoint on D, then ρ exponentiates to a unitary representation of G˜ (with additional remarks similar to those in Corollary 5.43). §5.8. Irreducible representations of SU (2) §5.9. Irreducible representations of compact Lie groups See e.g. Knapp (1988), Simon (1996), and Deitmar (2005), and innumerable other books. This material ultimately goes back to (E.)´ Cartan and Weyl. §5.10. Symmetry groups and projective representations See Varadarajan (1985), Tuynman & Wiegerinck (1987), Landsman (1998a), Cassinelli et al (2004), and Hall (2013). For different proofs of Theorem 5.59 (Bargmann, 1954) see Simms (1971) and Cassinelli et al (2004). Leaving out the anti-unitary symmetries is a pity; see e.g. Freed & Moore and Roberts (2016). §5.11. Position, momentum, and free Hamiltonian §5.12. Stone’s Theorem See Reed & Simon (1972), Schmudgen¨ (2012), Moretti (2013), Hall (2013), and many other books. Our proof of part 1 of Theorem 5.73 is original.